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 1-2 of 2 results.

A201950 Central coefficients in Product_{k=0..n-1} (1 + k*x + x^2).

Original entry on oeis.org

1, 0, 2, 6, 28, 160, 1078, 8358, 73260, 716112, 7721844, 91039740, 1164932470, 16077368580, 238037983558, 3763371442530, 63276351409092, 1127406030014112, 21218146474666864, 420611921077524912, 8759617763834095796, 191208185756772875880
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Examples

			The coefficients in Product_{k=0..n-1} (1+k*x+x^2) form triangle A201949:
(1);
1,(0), 1;
1, 1,(2), 1, 1;
1, 3, 5, (6), 5, 3, 1;
1, 6, 15, 24, (28), 24, 15, 6, 1;
1, 10, 40, 90, 139, (160), 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, (1078), 945, 629, 300, 91, 15, 1;
1, 21, 182, 861, 2520, 5019, 7377, (8358), 7377, 5019, 2520, 861, 182, 21, 1;
1, 28, 330, 2156, 8729, 23520, 45030, 65016, (73260), 65016, 45030, 23520, 8729, 2156, 330, 28, 1; ...
where coefficients in parenthesis form the initial terms of this sequence.

Links

Crossrefs

Cf. A201949, A201951, A201952, A201953.
Cf. A086672, A324304 (variant).

Programs

  • Mathematica
    Flatten[{1,Table[Coefficient[Expand[Product[1 + k*x + x^2,{k,0,n-1}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)
  • PARI
    {a(n) = polcoeff( prod(k=1,n,1+(k-1)*x+x^2+x*O(x^n)), n)}
for(n=0,30, print1(a(n),", "))
  • PARI
    /* From series BesselI(0, 2*log(1 - x)), after Ilya Gutkovskiy */
{a(n) = n!*polcoeff( sum(m=0,n, log(1 - x +x*O(x^n))^(2*m)/m!^2), n)}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 24 2019

Formula

Central terms of rows in irregular triangle A201949.
a(n) = (n-1)*a(n-1) + 2*A201952(n-1) for n>0. [corrected by Vaclav Kotesovec, May 04 2024]
E.g.f.: BesselI(0, 2*log(1 - x)). - Ilya Gutkovskiy, Feb 22 2019
E.g.f.: Sum_{n>=0} log(1 - x)^(2*n) / n!^2. [After Ilya Gutkovskiy - Paul D. Hanna, Feb 24 2019]

A324305 Triangle, read by rows, where the g.f. of row n equals Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a single '1' in row 1.

Original entry on oeis.org

1, 2, 0, 2, 9, 3, 18, 3, 9, 64, 48, 200, 96, 200, 48, 64, 625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625, 7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776, 117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649, 2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152
Offset: 1

Views

Author

Paul D. Hanna, Feb 28 2019

Keywords

Examples

			E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2) and satisfies A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
Explicitly,
A(x,y) = x + (2*y^2 + 2)*x^2/2! + (9*y^4 + 3*y^3 + 18*y^2 + 3*y + 9)*x^3/3! + (64*y^6 + 48*y^5 + 200*y^4 + 96*y^3 + 200*y^2 + 48*y + 64)*x^4/4! + (625*y^8 + 750*y^7 + 2775*y^6 + 2280*y^5 + 4300*y^4 + 2280*y^3 + 2775*y^2 + 750*y + 625)*x^5/5! + (7776*y^10 + 12960*y^9 + 46440*y^8 + 53640*y^7 + 100584*y^6 + 81360*y^5 + 100584*y^4 + 53640*y^3 + 46440*y^2 + 12960*y + 7776)*x^6/6! + (117649*y^12 + 252105*y^11 + 909979*y^10 + 1337700*y^9 + 2594501*y^8 + 2753415*y^7 + 3604342*y^6 + 2753415*y^5 + 2594501*y^4 + 1337700*y^3 + 909979*y^2 + 252105*y + 117649)*x^7/7! + ...
Setting y = 1 yields an o.g.f. of A006013:
A(x,y=1) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + 120175*x^9 + ... + binomial(3*n-2,n-1)/n * x^n + ...
TRIANGLE.
This triangle of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649;
2097152, 5505024, 20414464, 36040704, 73543680, 94730496, 133244544, 128389632, 133244544, 94730496, 73543680, 36040704, 20414464, 5505024, 2097152; ...
RELATED SERIES.
The e.g.f. may be defined by A(x,y) = Series_Reversion( x/G(x,y) )
where G(x,y) is the e.g.f. of A201949 and equals
G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2)
so that
G(x,y) = 1 + (1 + y^2)*x + (1 + y + 2*y^2 + y^3 + y^4)*x^2/2! + (1 + 3*y + 5*y^2 + 6*y^3 + 5*y^4 + 3*y^5 + y^6)*x^3/3! + (1 + 6*y + 15*y^2 + 24*y^3 + 28*y^4 + 24*y^5 + 15*y^6 + 6*y^7 + y^8)*x^4/4! + (1 + 10*y + 40*y^2 + 90*y^3 + 139*y^4 + 160*y^5 + 139*y^6 + 90*y^7 + 40*y^8 + 10*y^9 + y^10)*x^5/5! + ...
and G(x,y) = x / Series_Reversion( A(x,y) ).
RELATED TRIANGLE.
Triangle A201949 of coefficients in G(x,y) such that A(x/G(x,y),y) = x begins
1;
1, 0, 1;
1, 1, 2, 1, 1;
1, 3, 5, 6, 5, 3, 1;
1, 6, 15, 24, 28, 24, 15, 6, 1;
1, 10, 40, 90, 139, 160, 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, 1078, 945, 629, 300, 91, 15, 1; ...
where the g.f. of row n is Product_{k=0..n-1} (1 + k*y + y^2) for n >= 0.

Links

Crossrefs

Cf. A201949, A324304.
Cf. A324956, A324958.

Programs

  • PARI
    {T(n, k)=polcoeff(prod(j=0, n-2,  n + j*y + n*y^2), k, y)}
{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}
  • PARI
    /* A(x,y) = Series_Reversion(x/G(x,y)) where G(x,y) = e.g.f. A201949 */
{T(n,k) = my(G=1,A=x);
G = sum(m=0,n, x^m/m! * prod(j=0,m-1, 1 + j*y + y^2) +x*O(x^n));
A = serreverse(x/G);
n!*polcoeff(polcoeff(A,n,x),k,y)}
{for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))}

Formula

GENERATING FUNCTIONS.
E.g.f.: A(x,y) = x/(1 - y*A(x,y))^(1/y + y).
E.g.f.: A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y + y) ), wrt x.
E.g.f.: A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + k*y + n*y^2)
E.g.f.: A(x,y) = Series_Reversion( x/G(x,y) ) such that A(x/G(x,y),y) = x, where G(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + k*y + y^2) is the e.g.f. of A201949.
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)^2, where G = 1 + x*G(x)^3 is the g.f. of A001764.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (3*n-2)!/(2*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
T(n,n-1) = A324304(n) for n >= 1.
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