Search: id:A094665
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%I A094665
%S A094665 1,0,1,0,1,3,0,4,15,15,0,34,147,210,105,0,496,2370,4095,3150,945,0,
%T A094665 11056,56958,111705,107415,51975,10395,0,349504,191100,4114110,4579575,
%U A094665 2837835,945945,135135,0,14873104,85389132,197722980,244909665
%N A094665 Another version of triangular array in A083061 : triangle T(n,k), 0<=k<=n, 
               read by rows; given by [0, 1, 3, 6, 10, 15, 21, 28, ...] DELTA [1, 
               2, 3, 4, 5, 6, 7, 8, ...] where DELTA is the operator defined in 
               A084938.
%C A094665 Diagonals : A000007, A002105; A001147, A001880
%C A094665 Define polynomials P(n,x) = x(2x+1)P(n-1,x+1) - 2x^2P(n-1,x), P(0,x) 
               = 1. Sequence gives triangle read by rows, defined by P(n,x) = Sum_{k 
               = 0..n} T(n,k)*x^k . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jun 20 2004
%C A094665 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%C A094665 In A160464 we defined the coefficients of the ES1 matrix by ES1[2*m-1,
               n=1] = 2*eta(2*m-1) and the recurrence relation ES1[2*m-1,n] = ((2*n-2)/
               (2*n-1))*(ES1[2*m-1,n-1] - ES1[2*m-3,n-1]/(n-1)^2) for m the positive 
               and negative integers and n = 1, 2, 3, .. . As usual eta(m) = (1-2^(1-m))*zeta(m) 
               with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta 
               function. It is well-known that ES1[1-2*m,n=1] = (4^m-1)*(-bernoulli(2*m))/
               m for m = 1, 2, .. and together with the recurrence relation this 
               leads to ES1[-1,n] = 0.5 for n = 1, 2, .. .
%C A094665 We discovered that the nth term of the row coefficients ES1[1-2*m,n] 
               for m = 1, 2, 3, .. , can be generated with the rather simple polynomials 
               RES1(1-2*m,n) = (-1)^(m+1)*ECGP(1-2*m, n)/2^m. This discovery was 
               enabled by the recurrence relation for the RES1(1-2*m,n) which we 
               derived from the recurrence relation for the ES1[2*m-1,n] coefficients 
               and the fact that RES1(-1,n) = 0.5. The coefficients of the ECGP(1-2*m,
               n) polynomials led to this triangle and subsequently to triangle 
               A083061.
%C A094665 (End)
%H A094665 H.J.H. Tuenter, <a href="http://arXiv.org/abs/math.NT/0606080">Walking 
               into an absolute sum</a>
%F A094665 Sum_{k = 0..n} T(n, k) = A002105(n+1) . Sum_{k = 0..n} T(n, k)*2^(n-k) 
               = A000364(n); Euler numbers . Sum_{k = 0..n} T(n, k)*(-2)^(n-k) = 
               1.
%F A094665 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%F A094665 RES1(1-2*m,n) = n^2*RES1(3-2*m,n)-n*(2*n+1)*RES1(3-2*m,n+1)/2 for m = 
               2, 3, .. , with RES1(-1,n) = 0.5 for n = 1, 2, .. .
%F A094665 (End)
%e A094665 Triangle begins:
%e A094665 .1;
%e A094665 .0, 1;
%e A094665 .0, 1, 3;
%e A094665 .0, 4, 15, 15;
%e A094665 .0, 34, 147, 210, 105;
%e A094665 .0, 496, 2370, 4095, 3150, 945;
%e A094665 .0, 11056, 56958, 111705, 107415, 51975, 10395;
%e A094665 .0, 349504, 191100, 4114110, 4579575, 2837835, 945945, 135135;
%e A094665 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%e A094665 The first few ECGP(1-2*m,n) polynomials are: ECGP(-1,n) = 1; ECGP(-3,
               n) = n; ECGP(-5,n) = n + 3*n^2; ECGP(-7,n) = 4*n + 15*n^2+ 15*n^3 
               .
%e A094665 The first few RES1(1-2*m,n) are: RES1(-1,n) = (1/2)*(1); RES1(-3,n) = 
               (-1/4)*(n); RES1(-5,n) = (1/8)*(n+3*n^2); RES1(-7,n) = (-1/16)*(4*n+15*n^2+15*n^3).
%e A094665 (End)
%p A094665 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 
               2009: (Start)
%p A094665 restart; nmax:=8; imax := nmax: i:=0: T1(0,x):=1: T1(0,x+1):=1: for i 
               from 1 to imax do T1(i,x):= expand((2*x+1)*(x+1)*T1(i-1,x+1)-2*x^2*T1(i-1,
               x)): dx:=degree(T1(i, x)): for k from 0 to dx do c(k):=coeff(T1(i,
               x), x, k) od: T1(i,x+1):=sum(c(j)*(x+1)^(j),j=0..dx): od: for i from 
               0 to imax do for j from 0 to i do A083061(i,j):=coeff(T1(i,x), x, 
               j) od: od: for n from 0 to nmax do for k from 0 to n do T(n+1,k+1) 
               := A083061(n,k) od: od: T(0,0):=1: for n from 1 to nmax do T(n,0):=0 
               od: t:=0: for n from 0 to nmax do for k from 0 to n do a(t):= T(n,
               k): t:= t+1: od: od: seq(a(n),n=0..t-1);
%p A094665 (End)
%Y A094665 Cf. A000364 A084938 A083061.
%Y A094665 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%Y A094665 Cf. A160464, A083061 and A160468.
%Y A094665 A001147, A001880, A160470, A160471 and A160472 are the first five right 
               hand columns.
%Y A094665 (End)
%Y A094665 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 
               2009: (Start)
%Y A094665 Appears in A162005, A162006 and A162007.
%Y A094665 (End)
%Y A094665 Sequence in context: A079406 A068627 A074171 this_sequence A052439 A143073 
               A154725
%Y A094665 Adjacent sequences: A094662 A094663 A094664 this_sequence A094666 A094667 
               A094668
%K A094665 nonn,tabl
%O A094665 0,6
%A A094665 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 07 2004, Jun 12 2007

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