Search: id:A094816
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%I A094816
%S A094816 1,1,1,1,3,1,1,8,6,1,1,24,29,10,1,1,89,145,75,15,1,1,415,814,545,160,21,
%T A094816 1,1,2372,5243,4179,1575,301,28,1,1,16072,38618,34860,15659,3836,518,36,
%U A094816 1,1,125673,321690,318926,163191,47775,8274,834,45,1,1,1112083,2995011
%N A094816 Triangle read by rows: T(n,k), 0<=k<=n, = coefficients of Charlier polynomials 
               : A046716 transposed.
%C A094816 The a-sequence for this Sheffer matrix is A027641(n)/A027642(n) (Bernoulli 
               numbers) and the z-sequence is A130189(n)/ A130190(n). See the W. 
               Lang link.
%C A094816 Take the lower triangular matrix in A049020 and invert it, then read 
               by rows! - N. J. A. Sloane (njas(AT)research.att.com), Feb 07 2009
%C A094816 Exponential Riordan array [exp(x), ln(1/(1-x))]. Equal to A007318*A132393. 
               [From Paul Barry (pbarry(AT)wit.ie), Apr 23 2009]
%D A094816 W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite 
               modulus I, Acta Arith., 35 (1979), 1-16. [From N. J. A. Sloane (njas(AT)research.att.com), 
               Feb 07 2009]
%H A094816 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A094816.text">
               First 10 rows and more.</a>
%F A094816 E.g.f.: exp(t)/(1-t)^x = Sum_{n>=0} C(x,n)*t^n/n!. 1; 1, 1; 1, 3, 1; 
               1, 8, 6, 1; 1, 24, 29, 10, 1; ...
%F A094816 Sum_{k = 0..n} T(n, k)*x^k = C(x, n), Charlier polynomials; C(x, n)= 
               A000522(n), A001339(n), A082030(n) for x = 1, 2, 3 respectively.
%F A094816 T(n+1, k) = (n+1)*T(n, k) + T(n, k-1) - n*T(n-1, k) with T(0, 0) = 1, 
               T(0, k) = 0 if k>0, T(n, k) = 0 if k<0.
%F A094816 PS*A008275*PS as infinite lower triangular matrices, where PS is a triangle 
               with PS[n,k] = (-1)^k*A007318[n,k]. PS = 1/PS. [From Gerald McGarvey 
               (gerald.mcgarvey(AT)comcast.net), Aug 20 2009]
%e A094816 Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 23 2009: (Start)
%e A094816 Triangle begins
%e A094816 1,
%e A094816 1, 1,
%e A094816 1, 3, 1,
%e A094816 1, 8, 6, 1,
%e A094816 1, 24, 29, 10, 1,
%e A094816 1, 89, 145, 75, 15, 1,
%e A094816 1, 415, 814, 545, 160, 21, 1,
%e A094816 1, 2372, 5243, 4179, 1575, 301, 28, 1,
%e A094816 1, 16072, 38618, 34860, 15659, 3836, 518, 36, 1
%e A094816 Production matrix is
%e A094816 1, 1,
%e A094816 0, 2, 1,
%e A094816 0, 1, 3, 1,
%e A094816 0, 1, 3, 4, 1,
%e A094816 0, 1, 4, 6, 5, 1,
%e A094816 0, 1, 5, 10, 10, 6, 1,
%e A094816 0, 1, 6, 15, 20, 15, 7, 1,
%e A094816 0, 1, 7, 21, 35, 35, 21, 8, 1,
%e A094816 0, 1, 8, 28, 56, 70, 56, 28, 9, 1 (End)
%o A094816 (PARI) {T(n, k)= local(A); if(k<0|k>n, 0, A=x*O(x^n); polcoeff( n!*polcoeff( 
               exp(x+A)/(1-x+A)^y, n), k))} /* Michael Somos Nov 19 2006 */
%Y A094816 Diagonals : A000012, A002104; A000012, A000217.
%Y A094816 Row sums A000522, alternating row sums A024000.
%Y A094816 Sequence in context: A091698 A134380 A124469 this_sequence A097712 A157210 
               A034801
%Y A094816 Adjacent sequences: A094813 A094814 A094815 this_sequence A094817 A094818 
               A094819
%K A094816 nonn,tabl
%O A094816 0,5
%A A094816 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 12 2004

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