A202474 -id:A202474 - cp's OEIS frontend

A202476 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x + x^2).

1, 1, 2, 5, 11, 28, 74, 206, 601, 1826, 5766, 18851, 63676, 221678, 793958, 2920292, 11014653, 42543773, 168074091, 678403932, 2794920078, 11742254750, 50266213000, 219085792538, 971543475593, 4380664101448, 20071848941411, 93403455862117, 441206005123701

Offset: 0

Paul D. Hanna, Dec 19 2011

nonn

			The coefficients in Product_{k=1..n} (1+k*x+x^2), n>=0, form the triangle:
[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, 285, 539, 660, 539, 285, 90, 15, 1];
[1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1];
[1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
the antidiagonal sums of which form this sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A201951, A201826, A202474.

{a(n)=sum(k=0,n,polcoeff(prod(j=1,n-k,1+j*x+x^2),k))}

{a(n)=local(CF=1+x+x*O(x^n)); for(k=1, n-1, CF=(1+(n-k)*x+x^2)/(1 + x*(1+(n-k)*x+x^2) - x*CF+x*O(x^n))); polcoeff(1/(1-x*CF), n)}

Antidiagonal sums of the irregular triangle in which row n is defined by the g.f.: Product_{k=1..n} (1 + k*x + x^2) for n>=0.

G.f.: 1/(1 - x*(1+x+x^2)/(1 + x*(1+x+x^2) - x*(1+2*x+x^2)/(1 + x*(1+2*x+x^2) - x*(1+3*x+x^2)/(1 + x*(1+3*x+x^2) - x*(1+4*x+x^2)/(1 + x*(1+4*x+x^2) -...))))), a continued fraction.

A249790 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.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Nov 05 2014

			Triangle begins:
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, 285, 539, 660, 539, 285, 90, 15, 1;
1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 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, 358056, 216166, 90720, 25753, 4788, 554, 36, 1;
1, 45, 879, 9810, 69399, 327285, 1058399, 2394270, 3860922, 4516380, 3860922, 2394270, 1058399, 327285, 69399, 9810, 879, 45, 1;
1, 55, 1330, 18645, 168378, 1031085, 4400648, 13305545, 28862021, 45519870, 52885644, 45519870, 28862021, 13305545, 4400648, 1031085, 168378, 18645, 1330, 55, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1088, listing terms in rows 0..32 of flattened triangle.

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

Cf. A201949 (variant), A324956.

{T(n,k)=polcoeff(prod(m=1, n, 1 + m*x + x^2 +x*O(x^k)), k,x)}
for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))

E.g.f.: 1/(1 - x*y)^(1/y + 1 + y). - Paul D. Hanna, Mar 02 2019
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
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
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

A201826 Central coefficients in Product_{k=1..n} (1 + k*y + y^2).

Original entry on oeis.org

1, 1, 4, 18, 100, 660, 5038, 43624, 422252, 4516380, 52885644, 672781824, 9238314358, 136175455234, 2144494356834, 35930786795040, 638168940129732, 11976278012219556, 236791150694618872, 4919643784275283480, 107152493449339765396, 2441410548192907949196

Offset: 0

Paul D. Hanna, Dec 19 2011

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 100*x^4/4! + 660*x^5/5! + 5038*x^6/6! + 43624*x^7/7! + 422252*x^8/8! + 4516380*x^9/9! + 52885644*x^10/10! + ...
The coefficients in Product_{k=1..n} (1 + k*y + y^2), n>=0, form triangle A249790:
[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, 285, 539, 660, 539, 285, 90, 15, 1];
[1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1];
[1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
in which the central terms of the rows form this sequence.

Cf. A249790, A202474, A201950, A202476.

Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,1,n}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

{a(n) = polcoeff(prod(k=1,n, 1 + k*x + x^2 +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = n!*polcoeff( sum(m=0, n, log(1 - x +x*O(x^n))^(2*m)/m!^2 ) / (1 - x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 02 2019

E.g.f.: 1/(1-x) * Sum_{n>=0} log(1 - x)^(2*n) / n!^2. - Paul D. Hanna, Mar 02 2019

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A202476 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x + x^2).

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A249790 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.

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A201826 Central coefficients in Product_{k=1..n} (1 + k*y + y^2).

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