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.

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

Views

Author

Paul D. Hanna, Nov 05 2014

Keywords

Examples

			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; ...

Links

Crossrefs

Cf. A201826 (central coefficients), A202474 (a diagonal), A202476, A001710 (row sums).
Cf. A201949 (variant), A324956.

Programs

  • PARI
    {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(""))

Formula

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

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.

A324957 a(n) is the coefficient of y^(n-1) in Product_{k=0..n-2} (n + (n + k)*y + n*y^2), for n >= 1.

Original entry on oeis.org

1, 2, 30, 600, 17980, 701280, 33904822, 1956209024, 131229583596, 10037690006400, 862338700678884, 82216457762199552, 8614684257283258102, 984021646462806049280, 121700458370370286320750, 16202120194768984283381760, 2310243113824870018738497732, 351271199659878407029587277824, 56734597035765569017356665543968, 9700307843128908988513920450560000
Offset: 1

Views

Author

Paul D. Hanna, Mar 21 2019

Keywords

Comments

a(n) = A324956(n,n-1) for n >= 1.

Crossrefs

Cf. A324956.

Programs

  • PARI
    {a(n) = polcoeff( prod(k=0, n-2, n + (n+k)*y + n*y^2 +y*O(y^n)), n-1, y)}
for(n=1, 25, print1(a(n), ", "))
Showing 1-3 of 3 results.

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