Search: id:A028246
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%I A028246
%S A028246 1,1,1,1,3,2,1,7,12,6,1,15,50,60,24,1,31,180,390,360,120,1,63,602,2100,
%T A028246 3360,2520,720,1,127,1932,10206,25200,31920,20160,5040,1,255,6050,
%U A028246 46620,166824,317520,332640,181440,40320,1,511,18660,204630,1020600
%N A028246 Triangular array of numbers a(n,k) = Sum_{i=0..k} (-1)^(k-i)*C(k,i)*i^n; 
               n >= 1, 1<=k<=n.
%C A028246 Let M = n X n matrix with (i,j)-th entry a(n+1-j, n+1-i), e.g. if n = 
               3, M = [1 1 1; 3 1 0; 2 0 0]. Given a sequence s = [s(0)..s(n-1)], 
               let b = [b(0)..b(n-1)] be its inverse binomial transform and let 
               c = [c(0)..c(n-1)] = M^(-1)*transpose(b). Then s(k) = Sum_{i=0..n-1} 
               b(i)*binomial(k,i) = Sum_{i=0..n-1} c(i)*k^i, k=0..n-1. - Gary W. 
               Adamson, Nov 11, 2001.
%C A028246 Julius Worpitzky's 1883 algorithm generates Bernoulli numbers. By way 
               of example [Wikipedia]: B0 = 1 B1 = 1/1 - 1/2 B2 = 1/1 - 3/2 + 2/
               3 B3 = 1/1 - 7/2 + 12/3 - 6/4 B4 = 1/1 - 15/2 + 50/3 - 60/4 + 24/
               5 B5 = 1/1 - 31/2 + 180/3 - 390/4 + 360/5 - 120/6 B6 = 1/1 - 63/2 
               + 602/3 - 2100/4 + 3360/5 - 2520/6 + 720/7 ...(note that in this 
               algorithm odd n's for the Bernoulli numbers sum to 0, not 1 and the 
               sum for B1 = 1/2 = (1/1 - 1/2). B3 = 0 =(1 - 7/2 + 13/3 - 6/4) = 
               0. The summation for B4 = -1/30. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 09 2008]
%C A028246 Contribution from Tom Copeland (tcjpn(AT)msn.com), Oct 23 2008: (Start)
%C A028246 G(x,t) = 1/ {1 + [1-exp(x t)]/t} = 1 + 1 x + (2 + t) x^2/2! + (6 + 6t 
               + t^2) x^3/3! + ...
%C A028246 gives row polynomials for A090582-- reverse f-polynomials for the permutohedra 
               (see A019538).
%C A028246 G(x,t-1) = 1 + 1 x + (1 + t) x^2 / 2! + (1 + 4t + t^2) x^3 / 3! + ...
%C A028246 gives row polynomials for A008292, the h-polynomials for permutohedra.
%C A028246 G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ...
%C A028246 gives row polynomials for A028246.
%C A028246 (End)
%C A028246 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%C A028246 The Worpitzky triangle seems to be an apt name for this triangle.
%C A028246 (End)
%D A028246 A. Riskin and D. Beckwith, Problem 10231, Amer. Math. Monthly, 102 (1995), 
               175-176.
%D A028246 Wikipedia (Bernoulli numbers). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 09 2008]
%H A028246 H. Hasse, <a href="http://www.digizeitschriften.de/contentserver/contentserver?command=docconvert&docid=45206\
               9">Ein Summierungsverfahren fuer die Riemannsche Zeta-Reihe.</a>
%H A028246 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A028246 E.g.f.: -ln(1-y*(exp(x)-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Sep 28 2003
%F A028246 a(n, k) = S2(n, k)*(k-1)! where S2(n, k) is a Stirling number of the 
               second kind (cf. A008277). Also a(n,k) = T(n,k)/k, where T(n, k) 
               = A019538.
%F A028246 Essentially same triangle as triangle [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 
               6, 0, 7, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA 
               is Deleham's operator defined in A084938, but the notation is different.
%F A028246 Sum of terms in n-th row = A000629(n) - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 30 2005
%F A028246 The row generating polynomials P(n, t) are given by P(1, t)=t, P(n+1, 
               t)=t(t+1)diff(P(n, t), t) for n>=1 (see the Riskin and Beckwith reference). 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 09 2005
%F A028246 Additional comments from Gottfried Helms, Jul 12 2006:
%F A028246 (Start) Delta-matrix as can be read from H. Hasse's proof of a connection
%F A028246 between the zeta-function and Bernoulli numbers (see link below).
%F A028246 Let P = lower triangular matrix with entries P[row,col] = binom(row,col)
%F A028246 Let J = unit matrix with alternating signs J[r,r]=(-1)^r
%F A028246 Let N(m) = column matrix with N(m)(r) = (r+1)^m, N(1)--> natural numbers
%F A028246 Let V = Vandermonde matrix with V[r,c] = (r+1)^c
%F A028246 V is then also N(0)||N(1)||N(2)||N(3)... (indices r,c always beginning 
               at 0)
%F A028246 Then Delta = P*J * V and B' = N(-1)' * Delta
%F A028246 where B is the column matrix of Bernoulli numbers and ' means transpose,
%F A028246 or for the single k'th Bernoulli number B_k with the appropriate column 
               of Delta
%F A028246 B_k = N(-1)' * Delta[ *,k ] = N(-1)' * P*J * N(k)
%F A028246 Using a single column instead of V and assuming infinite dimension
%F A028246 H. Hasse showed that in x = N(-1) * P*J * N(s),
%F A028246 where s can be any complex number and s*zeta(1-s) = x.
%F A028246 His theorem reads: s*zeta(1-s) = sum_{n=0..inf} ( (n+1)^-1 * delta(n,
               s) )
%F A028246 where delta(n,s) = sum_{j=0..n} [ (-1)^j * binom(n,j) * (j+1)^s ] (end)
%F A028246 The k-th row (k>=1) contains a(i, k) for i=1 to k, where a(i, k) satisfies 
               Sum_{i=1..n} C(i, 1)^k = 2 * C(n+1, 2) * Sum_{i=1..k} a(i, k) * C(n-1, 
               i-1)/(i+1). E.g. Row 3 contains 1, 3, 2 so Sum_{i=1..n} C(i, 1)^3 
               = 2 * C(n+1, 2) * [ a(1, 3)/2 +a(2, 3) *C(n-1, 1)/3 +a(3, 3)*C(n-1, 
               2)/4 ] = [ (n+1)*n ] * [ 1/2 +(3/3)*C(n-1, 1) +(2/4)*C(n-1, 2) ] 
               = ( n^2 +n ) * ( n -1 +[ C(n-1, 2) +1 ]/2 ) = C(n+1, 2)^2. See A000537 
               for more details ( 1^3 +2^3 +3^3 +4^3 +5^3 +... ). - Andre F. Labossiere 
               (boronali(AT)laposte.net), Sep 22 2003
%F A028246 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%F A028246 a(n,k) = k*a(n-1,k) + (k-1)*a(n-1,k-1) with a(n,1) = 1 and a(n,n) = (n-1)!
%F A028246 (End)
%e A028246 1; 1,1; 1,3,2; 1,7,12,6; 1,15,50,60,24; ...
%e A028246 Row 5 of triangle is {1,15,50,60,24}, which is {1,15,25,10,1} times {0!,
               1!,2!,3!,4!}.
%p A028246 a:=(n,k)->add( (-1)^(k-i)*C(k,i)*i^n,i=0..k)/k;
%o A028246 (PARI) T(n,k)=if(k<0|k>n,0,n!*polcoeff((x/log(1+x+x^2*O(x^n)))^(n+1),
               n-k))
%Y A028246 Dropping the column of 1's gives A053440. See also A008277.
%Y A028246 Without the k in the denominator (in the definition), we get A019538. 
               See also the Stirling number triangle A008277.
%Y A028246 Cf. A087127, A087107, A087108, A087109, A087110, A087111, A084938 A075263.
%Y A028246 Row sums give A000629.
%Y A028246 A027642, A002445 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 09 
               2008]
%Y A028246 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%Y A028246 Appears in A161739 (RSEG2 triangle), A161742 and A161743.
%Y A028246 (End)
%Y A028246 Sequence in context: A056151 A134436 A163626 this_sequence A082038 A143774 
               A158474
%Y A028246 Adjacent sequences: A028243 A028244 A028245 this_sequence A028247 A028248 
               A028249
%K A028246 nonn,easy,nice,tabl
%O A028246 1,5
%A A028246 N. J. A. Sloane (njas(AT)research.att.com), Doug McKenzie (mckfam4(AT)aol.com)
%E A028246 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 14 2000
%E A028246 Definition corrected by Li Guo, Dec 16 2006
%E A028246 Link to the Hasse paper repaired Peter Luschny (peter(AT)luschny.de), 
               Apr 21 2009
%E A028246 Typo in link corrected by Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Oct 17 2009

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