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 1-7 of 7 results.

A124546 Row 6 of rectangular table A124540; equals the self-convolution 6th power of A124536 (row 6 of table A124530).

Original entry on oeis.org

1, 6, 57, 470, 3756, 29658, 233241, 1836912, 14543877, 116087596, 936035298, 7636193394, 63106764294, 528842660346, 4497737044197, 38849799300246, 341016182672523, 3043519729680600, 27629723055323671, 255224042883932790
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124540, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k*R_k(y)^n ]^n for n>=0.

Crossrefs

Cf. A124535; A124540 (table); other rows: A124531, A124542, A124543, A124544, A124545.

Programs

  • PARI
    {a(n)=local(R);R=vector(n+7,r,vector(n+7,c,1)); for(i=0,n+6,for(r=0,n+6,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); Vec(Ser(R[7])^6+O(x^(n+1)))[n+1]}

Formula

G.f.: A(x) = [ Sum_{n>=0} x^n*R_n(x)^6 ]^6, where R_n(x) is the g.f. of row n in table A124540.

A124530 Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^(n*k) for n>=0, with R_0(y) = 1/(1-y).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 10, 16, 1, 1, 1, 5, 16, 39, 62, 1, 1, 1, 6, 23, 71, 174, 274, 1, 1, 1, 7, 31, 113, 351, 858, 1332, 1, 1, 1, 8, 40, 166, 608, 1891, 4564, 6978, 1, 1, 1, 9, 50, 231, 961, 3535, 10888, 25793, 38873, 1, 1, 1, 10, 61, 309, 1427
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

See table A124540, in which row n equals the n-th self-convolution of row n of A124530 (this table).

Examples

			Row g.f.s R_n(y) simultaneously satisfy:
R_n(y) = 1 + y*R_1(y)^n + y^2*R_2(y)^(2n) + y^3*R_3(y)^(3n) +...
more explicitly:
R_0 = 1 + y + y^2 + y^3 + y^4 + ...
R_1 = 1 + y*(R_1)^1 + y^2*(R_2)^2 + y^3*(R_3)^3 + y^4*(R_4)^4 + ...
R_2 = 1 + y*(R_1)^2 + y^2*(R_2)^4 + y^3*(R_3)^6 + y^4*(R_4)^8 +...
R_3 = 1 + y*(R_1)^3 + y^2*(R_2)^6 + y^3*(R_3)^9 + y^4*(R_4)^12 +...
R_4 = 1 + y*(R_1)^4 + y^2*(R_2)^8 + y^3*(R_3)^12 + y^4*(R_4)^16 +...
etc., for all rows.
Rectangular table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,1,2,5,16,62,274,1332,6978,38873,228090,1399625,8933506,...
1,1,3,10,39,174,858,4564,25793,153301,951325,6130757,40861658,...
1,1,4,16,71,351,1891,10888,66139,420235,2775981,18978873,...
1,1,5,23,113,608,3535,21844,141809,959882,6738850,48877221,...
1,1,6,31,166,961,5977,39363,271564,1949165,14487241,111115804,...
1,1,7,40,231,1427,9430,65810,480077,3637345,28502254,230271472,...
1,1,8,50,309,2024,14134,104028,798954,6363948,52370770,443997440,...
1,1,9,61,401,2771,20357,157383,1267833,10579140,91111871,...
1,1,10,73,508,3688,28396,229810,1935562,16866694,151563677,...
1,1,11,86,631,4796,38578,325860,2861457,25969694,242836861,...
1,1,12,100,771,6117,51261,450748,4116641,38819122,376841378,...

Crossrefs

Rows: A124531, A124532, A124533, A124534, A124535, A124536; A124537 (diagonal), A124538 (antidiagonal sums); related tables: A124539, A124540; A124460 (variant).

Programs

  • PARI
    T(n,k)=local(m=max(n,k),R);R=vector(m+1,r,vector(m+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,m, for(r=0,m, R[r+1]=Vec(sum(c=0,m, x^c*Ser(R[c+1])^(r*c)+O(x^(m+1)))))); R[n+1][k+1]

Formula

G.f.: A(x,y) = Sum_{n>=0} x^n*R_n(y) = Sum_{k>=0} y^k/(1 - x*R_k(y)^k).

A124531 Row 1 of rectangular table A124530.

Original entry on oeis.org

1, 1, 2, 5, 16, 62, 274, 1332, 6978, 38873, 228090, 1399625, 8933506, 59066261, 403241292, 2835267821, 20490128048, 151951503074, 1154770194362, 8983396031267, 71473650532630, 581142591346325, 4825842125683150
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Crossrefs

Cf. A124530 (table); other rows: A124532, A124533, A124534, A124535, A124536.

Programs

  • PARI
    a(n)=local(R);R=vector(n+2,r,vector(n+2,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); R[2][n+1]

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^n, where R_n(x) is the g.f. of row n in table A124530.

A124535 Row 5 of rectangular table A124530.

Original entry on oeis.org

1, 1, 6, 36, 216, 961, 5977, 39363, 271564, 1949165, 14487241, 111115804, 877086405, 7109569724, 59075905996, 502464618671, 4369068613929, 38796820526316, 351496911119002, 3246426019892427, 30543990287835441
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Crossrefs

Cf. A124530 (table); other rows: A124531, A124532, A124533, A124534, A124536.

Programs

  • PARI
    a(n)=local(R);R=vector(n+6,r,vector(n+6,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); R[6][n+1]

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^(5n), where R_n(x) is the g.f. of row n in table A124530.

A124532 Row 2 of rectangular table A124530.

Original entry on oeis.org

1, 1, 3, 10, 39, 174, 858, 4564, 25793, 153301, 951325, 6130757, 40861658, 280767621, 1983859580, 14385651988, 106878699675, 812480791324, 6312686006725, 50083418434737, 405430892640225, 3346568909331984
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Crossrefs

Cf. A124530 (table); other rows: A124531, A124533, A124534, A124535, A124536.

Programs

  • PARI
    a(n)=local(R);R=vector(n+3,r,vector(n+3,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); R[3][n+1]

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^(2n), where R_n(x) is the g.f. of row n in table A124530.

A124533 Row 3 of rectangular table A124530.

Original entry on oeis.org

1, 1, 4, 16, 71, 351, 1891, 10888, 66139, 420235, 2775981, 18978873, 133828922, 970678790, 7226115267, 55115404005, 430080085093, 3429311454089, 27912555377062, 231710034354364, 1960247357996533, 16889105788701591
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Crossrefs

Cf. A124530 (table); other rows: A124531, A124532, A124534, A124535, A124536.

Programs

  • PARI
    a(n)=local(R);R=vector(n+4,r,vector(n+4,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); R[4][n+1]

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^(3n), where R_n(x) is the g.f. of row n in table A124530.

A124534 Row 4 of rectangular table A124530.

Original entry on oeis.org

1, 1, 5, 25, 113, 608, 3535, 21844, 141809, 959882, 6738850, 48877221, 365145267, 2803002587, 22066904802, 177881536038, 1466278969213, 12345543329079, 106069531868783, 929158597546721, 8292429187449462, 75348425058995464
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Crossrefs

Cf. A124530 (table); other rows: A124531, A124532, A124533, A124535, A124536.

Programs

  • PARI
    a(n)=local(R);R=vector(n+5,r,vector(n+5,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); R[5][n+1]

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^(4n), where R_n(x) is the g.f. of row n in table A124530.
Showing 1-7 of 7 results.

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