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 1-10 of 13 results. Next

A124541 G.f.: A(x) = R_2(x)/R_1(x), where R_2(x) and R_1(x) are the g.f.s of row 2 (A124542) and row 1 (A124531), respectively, of table A124540.

Original entry on oeis.org

1, 1, 4, 15, 63, 295, 1502, 8167, 46873, 281672, 1761798, 11418480, 76415644, 526594846, 3728435747, 27073765165, 201325681384, 1531247489953, 11899881220174, 94409837555587, 764105555574024, 6304959856949278
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.

Examples

			G.f.: A(x) = R_2(x)/R_1(x), where row g.f.s are:
R_2(x) = 1 + 2x + 7x^2 + 26x^3 + 107x^4 + 486x^5 + 2398x^6 + ... and
R_1(x) = 1 + x + 2x^2 + 5x^3 + 16x^4 + 62x^5 + 274x^6 + ..., so that
A(x) = 1 + x + 4*x^2 + 15*x^3 + 63*x^4 + 295*x^5 + 1502*x^6 + ...

Crossrefs

Cf. A124540 (table); rows: A124531, A124542, A124543, A124544, A124545, A124546.

Programs

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

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).

A124540 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 ]^n for n>=0, with R_0(y) = 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 26, 16, 0, 1, 5, 26, 73, 107, 62, 0, 1, 6, 40, 156, 369, 486, 274, 0, 1, 7, 57, 285, 939, 1959, 2398, 1332, 0, 1, 8, 77, 470, 1995, 5764, 10912, 12668, 6978, 0, 1, 9, 100, 721, 3756, 13976, 36248, 63543, 70863, 38873, 0
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2006

Keywords

Comments

Antidiagonal sums equal row 1 (A124531).

Examples

			Row g.f.s R_n(y) simultaneously satisfy:
R_n(y) = [1 + y*R_1(y)^n + y^2*R_2(y)^n + y^3*R_3(y)^n +...]^n
more explicitly:
R_0 = [1 + y + y^2 + y^3 + y^4 + ...]^0 = 1;
R_1 = [1 + y*(R_1)^1 + y^2*(R_2)^1 + y^3*(R_3)^1 + y^4*(R_4)^1 +...]^1;
R_2 = [1 + y*(R_1)^2 + y^2*(R_2)^2 + y^3*(R_3)^2 + y^4*(R_4)^2 +...]^2;
R_3 = [1 + y*(R_1)^3 + y^2*(R_2)^3 + y^3*(R_3)^3 + y^4*(R_4)^3 +...]^3;
R_4 = [1 + y*(R_1)^4 + y^2*(R_2)^4 + y^3*(R_3)^4 + y^4*(R_4)^4 +...]^4;
etc., for all rows.
Table begins:
1,0,0,0,0,0,0,0,0,0,0,...
1,1,2,5,16,62,274,1332,6978,38873,228090,...
1,2,7,26,107,486,2398,12668,70863,416304,2552490,...
1,3,15,73,369,1959,10912,63543,385341,2424988,15788469,...
1,4,26,156,939,5764,36248,233900,1549193,10529052,73390856,...
1,5,40,285,1995,13976,98665,704810,5107950,37619020,281850156,...
1,6,57,470,3756,29658,233241,1836912,14543877,116087596,936035298,...
1,7,77,721,6482,57057,495922,4282895,36922550,318834341,2765000007,...
1,8,100,1048,10474,101800,970628,9140344,85445683,795971176,7410928800,...
1,9,126,1461,16074,171090,1777416,18151272,183201255,1834958107,...
1,10,155,1970,23665,273902,3081700,33954660,368443380,3954149640,...
1,11,187,2585,33671,421179,5104528,60398327,701775756,8042277034,...
1,12,222,3316,46557,626028,8133916,102916452,1275653922,15559229828,...

Crossrefs

Rows: A124531, A124542, A124543, A124544, A124545, A124546; diagonals: A124547, A124548, A124549; related tables: A124530, A124550, A124460.

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)))))); Vec(Ser(R[n+1])^n+O(x^(k+1)))[k+1]

Formula

Let S_n(y) be the g.f. of row n in table A124530, then R_n(y) = S_n(y)^n and so S_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0, where R_n(y) is the g.f. of row n in this table.

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.

A124536 Row 6 of rectangular table A124530.

Original entry on oeis.org

1, 1, 7, 49, 343, 2401, 9430, 65810, 480077, 3637345, 28502254, 230271472, 1913354190, 16318251874, 142611810220, 1275372111383, 11657396456231, 108792892306605, 1035708759921763, 10049835806235885, 99321845447658052
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, A124535.

Programs

  • PARI
    a(n)=local(R);R=vector(n+7,r,vector(n+7,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[7][n+1]

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^(6n), where R_n(x) is the g.f. of row n in table A124530.

A124542 Row 2 of rectangular table A124540; equals the self-convolution of A124532 (row 2 of table A124530).

Original entry on oeis.org

1, 2, 7, 26, 107, 486, 2398, 12668, 70863, 416304, 2552490, 16254406, 107095090, 727834866, 5089682472, 36548625188, 269065010063, 2027942075946, 15630423416331, 123079853443384, 989356860469923, 8112792202324232
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. A124532; A124540 (table); other rows: A124531, A124543, A124544, A124545, A124546.

Programs

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

A124543 Row 3 of rectangular table A124540; equals the self-convolution cube of A124533 (row 3 of table A124530).

Original entry on oeis.org

1, 3, 15, 73, 369, 1959, 10912, 63543, 385341, 2424988, 15788469, 106075089, 733801709, 5217101283, 38060759175, 284533309380, 2177136417042, 17032924895739, 136129119703837, 1110507731328900, 9240322072954209
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. A124533; A124540 (table); other rows: A124531, A124542, A124544, A124545, A124546.

Programs

  • PARI
    {a(n)=local(R);R=vector(n+4,r,vector(n+4,c,1)); for(i=0,n+3,for(r=0,n+3,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); Vec(Ser(R[4])^3+O(x^(n+1)))[n+1]}

Formula

G.f.: A(x) = [ Sum_{n>=0} x^n*R_n(x)^3 ]^3, where R_n(x) is the g.f. of row n in table A124540.
Showing 1-10 of 13 results. Next

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