cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Paul D. Hanna, Dec 19 2011

Keywords

Examples

			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.

Crossrefs

Cf. A249790, A202474, A201950, A202476.

Programs

  • Mathematica
    Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,1,n}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)
  • PARI
    {a(n) = polcoeff(prod(k=1,n, 1 + k*x + x^2 +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {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

Formula

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

A324956 Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (n + k)*y + n*y^2), as read by rows of terms k = 0..2*n-2, for n >= 1.

Original entry on oeis.org

1, 2, 2, 2, 9, 21, 30, 21, 9, 64, 240, 488, 600, 488, 240, 64, 625, 3250, 8775, 15080, 17980, 15080, 8775, 3250, 625, 7776, 51840, 176040, 387360, 606384, 701280, 606384, 387360, 176040, 51840, 7776, 117649, 957999, 3935239, 10557540, 20437361, 29924601, 33904822, 29924601, 20437361, 10557540, 3935239, 957999, 117649, 2097152, 20185088, 97484800, 308768768, 711782400, 1258039552, 1753753728, 1956209024, 1753753728, 1258039552, 711782400, 308768768, 97484800, 20185088, 2097152
Offset: 1

Views

Author

Paul D. Hanna, Mar 21 2019

Keywords

Examples

			E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k starts
A(x) = x + (2*y^2 + 2*y + 2)*x^2/2! + (9*y^4 + 21*y^3 + 30*y^2 + 21*y + 9)*x^3/3! + (64*y^6 + 240*y^5 + 488*y^4 + 600*y^3 + 488*y^2 + 240*y + 64)*x^4/4! + (625*y^8 + 3250*y^7 + 8775*y^6 + 15080*y^5 + 17980*y^4 + 15080*y^3 + 8775*y^2 + 3250*y + 625)*x^5/5! + (7776*y^10 + 51840*y^9 + 176040*y^8 + 387360*y^7 + 606384*y^6 + 701280*y^5 + 606384*y^4 + 387360*y^3 + 176040*y^2 + 51840*y + 7776)*x^6/6! + (117649*y^12 + 957999*y^11 + 3935239*y^10 + 10557540*y^9 + 20437361*y^8 + 29924601*y^7 + 33904822*y^6 + 29924601*y^5 + 20437361*y^4 + 10557540*y^3 + 3935239*y^2 + 957999*y + 117649)*x^7/7! + (2097152*y^14 + 20185088*y^13 + 97484800*y^12 + 308768768*y^11 + 711782400*y^10 + 1258039552*y^9 + 1753753728*y^8 + 1956209024*y^7 + 1753753728*y^6 + 1258039552*y^5 + 711782400*y^4 + 308768768*y^3 + 97484800*y^2 + 20185088*y + 2097152)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins:
1;
2, 2, 2;
9, 21, 30, 21, 9;
64, 240, 488, 600, 488, 240, 64;
625, 3250, 8775, 15080, 17980, 15080, 8775, 3250, 625;
7776, 51840, 176040, 387360, 606384, 701280, 606384, 387360, 176040, 51840, 7776;
117649, 957999, 3935239, 10557540, 20437361, 29924601, 33904822, 29924601, 20437361, 10557540, 3935239, 957999, 117649;
2097152, 20185088, 97484800, 308768768, 711782400, 1258039552, 1753753728, 1956209024, 1753753728, 1258039552, 711782400, 308768768, 97484800, 20185088, 2097152; ...

Crossrefs

Cf. A324957.
Cf. A249790.

Programs

  • PARI
    {T(n, k) = polcoeff(prod(m=0, n-2, n + (n+m)*y + n*y^2 +y*O(y^k)), k, y)}
for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))
  • PARI
    {T(n,k) = my(A = serreverse( x*(1 - x*y +x*O(x^n) )^(1/y+1+y)));
n!*polcoeff(polcoeff(A,n,x),k,y)}
for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))

Formula

E.g.f. A(x) = Sum_{n>=1} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (n + k)*y + n*y^2).
(2) A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y+1+y) ).
(3) A(x,y) = x/(1 - y*A(x))^(1/y+1+y).
(4) A(x,y) = x*Sum_{n>=0} A(x,y)^n/n! * Product_{k=0..n-1} (1 + (k+1)*y + y^2).
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
E.g.f. at y = 1: A(x,y=1) = x*G(x)^3, where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (4*n-2)!/(3*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
T(n,n-1) = A324957(n) for n >= 1.

A324960 Triangle of coefficients T(n,k) of y^k in Product_{k=0..n-1} (1 + (k+2)*y + y^2), read by rows of terms k = 0..2*n, for n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 1, 5, 8, 5, 1, 1, 9, 29, 42, 29, 9, 1, 1, 14, 75, 196, 268, 196, 75, 14, 1, 1, 20, 160, 660, 1519, 2000, 1519, 660, 160, 20, 1, 1, 27, 301, 1800, 6299, 13293, 17038, 13293, 6299, 1800, 301, 27, 1, 1, 35, 518, 4235, 21000, 65485, 129681, 162890, 129681, 65485, 21000, 4235, 518, 35, 1, 1, 44, 834, 8932, 59633, 258720, 740046, 1395504, 1725372, 1395504, 740046, 258720, 59633, 8932, 834, 44, 1, 1, 54, 1275, 17316, 149787, 863982, 3386879, 9054684, 16420458, 20044728, 16420458, 9054684, 3386879, 863982, 149787, 17316, 1275, 54, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 20 2019

Keywords

Examples

			E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k starts
A(x,y) = 1 + (y^2 + 2*y + 1)*x + (y^4 + 5*y^3 + 8*y^2 + 5*y + 1)*x^2/2! + (y^6 + 9*y^5 + 29*y^4 + 42*y^3 + 29*y^2 + 9*y + 1)*x^3/3! + (y^8 + 14*y^7 + 75*y^6 + 196*y^5 + 268*y^4 + 196*y^3 + 75*y^2 + 14*y + 1)*x^4/4! + (y^10 + 20*y^9 + 160*y^8 + 660*y^7 + 1519*y^6 + 2000*y^5 + 1519*y^4 + 660*y^3 + 160*y^2 + 20*y + 1)*x^5/5! + (y^12 + 27*y^11 + 301*y^10 + 1800*y^9 + 6299*y^8 + 13293*y^7 + 17038*y^6 + 13293*y^5 + 6299*y^4 + 1800*y^3 + 301*y^2 + 27*y + 1)*x^6/6! + (y^14 + 35*y^13 + 518*y^12 + 4235*y^11 + 21000*y^10 + 65485*y^9 + 129681*y^8 + 162890*y^7 + 129681*y^6 + 65485*y^5 + 21000*y^4 + 4235*y^3 + 518*y^2 + 35*y + 1)*x^7/7! + (y^16 + 44*y^15 + 834*y^14 + 8932*y^13 + 59633*y^12 + 258720*y^11 + 740046*y^10 + 1395504*y^9 + 1725372*y^8 + 1395504*y^7 + 740046*y^6 + 258720*y^5 + 59633*y^4 + 8932*y^3 + 834*y^2 + 44*y + 1)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins:
  1;
  1, 2, 1;
  1, 5, 8, 5, 1;
  1, 9, 29, 42, 29, 9, 1;
  1, 14, 75, 196, 268, 196, 75, 14, 1;
  1, 20, 160, 660, 1519, 2000, 1519, 660, 160, 20, 1;
  1, 27, 301, 1800, 6299, 13293, 17038, 13293, 6299, 1800, 301, 27, 1;
  1, 35, 518, 4235, 21000, 65485, 129681, 162890, 129681, 65485, 21000, 4235, 518, 35, 1;
  1, 44, 834, 8932, 59633, 258720, 740046, 1395504, 1725372, 1395504, 740046, 258720, 59633, 8932, 834, 44, 1;
  1, 54, 1275, 17316, 149787, 863982, 3386879, 9054684, 16420458, 20044728, 16420458, 9054684, 3386879, 863982, 149787, 17316, 1275, 54, 1; ...

Crossrefs

Cf. A324961, A324962.
Cf. A324958, A249790.

Programs

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

Formula

E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
(2) A(x,y) = 1/(1 - x*y)^((1+y)^2/y).
(3) x = Sum_{n>=1} (x/A(x,y))^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2).
Row sums are (n+3)!/3! for row n >= 0.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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