A249790 - OEIS

%I #12 Mar 21 2019 01:00:17

%S 1,1,1,1,1,3,4,3,1,1,6,14,18,14,6,1,1,10,39,80,100,80,39,10,1,1,15,90,

%T 285,539,660,539,285,90,15,1,1,21,181,840,2339,4179,5038,4179,2339,

%U 840,181,21,1,1,28,329,2128,8400,21392,36630,43624,36630,21392,8400,2128,329,28,1,1,36,554,4788,25753,90720,216166,358056,422252

%N Triangle in which row n lists the coefficients in Product_{k=1..n} (1 + k*x + x^2), for n>=0, as read by rows.

%H Paul D. Hanna, <a href="/A249790/b249790.txt">Table of n, a(n) for n = 0..1088, listing terms in rows 0..32 of flattened triangle.</a>

%F E.g.f.: 1/(1 - x*y)^(1/y + 1 + y). - _Paul D. Hanna_, Mar 02 2019

%F E.g.f.: A(x,y) = 1/(1-x*y) * Sum_{k>=0} (1/y^k + y^k)/2^(0^k) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!). - _Paul D. Hanna_, Mar 02 2019

%F E.g.f. of diagonal k: (1/y^k)/(1-x*y) * Sum_{n>=0} (-log(1 - x*y))^(2*n+k) / (n!*(n+k)!) for k >= 0. - _Paul D. Hanna_, Mar 02 2019

%F E.g.f.: A(x,y) = x / Series_Reversion( F(x,y) ) such that F(x/A(x,y),y) = x, where F(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (n+k)*y + n*y^2). - _Paul D. Hanna_, Mar 02 2019

%e Triangle begins:

%e 1;

%e 1, 1, 1;

%e 1, 3, 4, 3, 1;

%e 1, 6, 14, 18, 14, 6, 1;

%e 1, 10, 39, 80, 100, 80, 39, 10, 1;

%e 1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1;

%e 1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1;

%e 1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1;

%e 1, 36, 554, 4788, 25753, 90720, 216166, 358056, 422252, 358056, 216166, 90720, 25753, 4788, 554, 36, 1;

%e 1, 45, 879, 9810, 69399, 327285, 1058399, 2394270, 3860922, 4516380, 3860922, 2394270, 1058399, 327285, 69399, 9810, 879, 45, 1;

%e 1, 55, 1330, 18645, 168378, 1031085, 4400648, 13305545, 28862021, 45519870, 52885644, 45519870, 28862021, 13305545, 4400648, 1031085, 168378, 18645, 1330, 55, 1; ...

%o (PARI) {T(n,k)=polcoeff(prod(m=1, n, 1 + m*x + x^2 +x*O(x^k)), k,x)}

%o for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))

%Y Cf. A201826 (central coefficients), A202474 (a diagonal), A202476, A001710 (row sums).

%Y Cf. A201949 (variant), A324956.

%K nonn,tabf

%O 0,6

%A _Paul D. Hanna_, Nov 05 2014