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 14 results. Next

A124461 Row 1 of rectangular table A124460; also equals the antidiagonal sums of table A124460.

Original entry on oeis.org

1, 2, 4, 9, 23, 66, 210, 731, 2744, 10959, 46058, 202028, 919386, 4321837, 20916763, 103958935, 529512396, 2759340117, 14690668955, 79813438230, 442057104246, 2493917931493, 14320972817368, 83652035642564, 496769855224824
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Comments

In table A124460, the o.g.f. of row n, R_n(y), satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.

Crossrefs

Cf. A124460 (table); other rows: A124462, A124463, A124464, A124465, A124466.

Programs

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

Formula

O.g.f.: A(x) = Sum_{n>=0} x^n*R_n(x) = Sum_{k>=0} x^k/(1 - x*R_k(x)), where R_n(x) is the o.g.f. of row n in table A124460.

A124462 Row 2 of rectangular table A124460.

Original entry on oeis.org

1, 3, 8, 23, 73, 251, 919, 3549, 14371, 60720, 266481, 1209807, 5662008, 27238884, 134391046, 678739990, 3503708942, 18462855900, 99211177417, 543161148837, 3027439667989, 17167987227428, 98995692542858, 580166879766649
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Crossrefs

Cf. A124460 (table); other rows: A124461, A124463, A124464, A124465, A124466.

Programs

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

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^2, where R_n(x) is the g.f. of row n in table A124460 and satisfies: R_n(x) = Sum_{k>=0} x^k * R_k(x)^n for n>=0.

A124463 Row 3 of rectangular table A124460.

Original entry on oeis.org

1, 4, 13, 44, 162, 637, 2622, 11188, 49293, 223768, 1044661, 5006126, 24580704, 123464593, 633467893, 3315948760, 17690431736, 96101620364, 531196784996, 2985589206506, 17053375055317, 98942675518256, 582858713958087
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Crossrefs

Cf. A124460 (table); other rows: A124461, A124462, A124464, A124465, A124466.

Programs

  • PARI
    {a(n)=local(m=max(n,3),R=vector(m+1,r,vector(m+1,c,binomial(r+c-2,c-1)))); for(i=0,m,for(r=0,m,R[r+1]=Vec(sum(c=0,m,x^c*Ser(R[c+1])^r+O(x^(m+1))))));R[4][n+1]}

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^3, where R_n(x) is the g.f. of row n in table A124460 and satisfies: R_n(x) = Sum_{k>=0} x^k * R_k(x)^n for n>=0.

A124464 Row 4 of rectangular table A124460.

Original entry on oeis.org

1, 5, 19, 73, 302, 1325, 6032, 28193, 134825, 659011, 3290110, 16764206, 87103106, 461090076, 2484768459, 13621130998, 75906831145, 429768775851, 2470872560536, 14418770507660, 85367306874021, 512604419523512
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Crossrefs

Cf. A124460 (table); other rows: A124461, A124462, A124463, A124465, A124466.

Programs

  • PARI
    {a(n)=local(m=max(n,4),R=vector(m+1,r,vector(m+1,c,binomial(r+c-2,c-1)))); for(i=0,m,for(r=0,m,R[r+1]=Vec(sum(c=0,m,x^c*Ser(R[c+1])^r+O(x^(m+1))))));R[5][n+1]}

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^4, where R_n(x) is the g.f. of row n in table A124460 and satisfies: R_n(x) = Sum_{k>=0} x^k * R_k(x)^n for n>=0.

A124465 Row 5 of rectangular table A124460.

Original entry on oeis.org

1, 6, 26, 111, 506, 2437, 12118, 61499, 317485, 1666371, 8891543, 48221602, 265708512, 1486905853, 8446728295, 48690214869, 284692228671, 1687855348715, 10143257898587, 61768997909283, 381059689323080, 2380852983417944
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Crossrefs

Cf. A124460 (table); other rows: A124461, A124462, A124463, A124464, A124466.

Programs

  • PARI
    {a(n)=local(m=max(n,5),R=vector(m+1,r,vector(m+1,c,binomial(r+c-2,c-1)))); for(i=0,m,for(r=0,m,R[r+1]=Vec(sum(c=0,m,x^c*Ser(R[c+1])^r+O(x^(m+1))))));R[6][n+1]}

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^5, where R_n(x) is the g.f. of row n in table A124460 and satisfies: R_n(x) = Sum_{k>=0} x^k * R_k(x)^n for n>=0.

A124466 Row 6 of rectangular table A124460.

Original entry on oeis.org

1, 7, 34, 159, 788, 4117, 22143, 121079, 670811, 3764758, 21408813, 123367344, 720303998, 4260491461, 25523584313, 154835572102, 950950450917, 5911757419002, 37193168793829, 236764641742504, 1524754216743136
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Crossrefs

Cf. A124460 (table); other rows: A124461, A124462, A124463, A124464, A124465.

Programs

  • PARI
    {a(n)=local(m=max(n,6),R=vector(m+1,r,vector(m+1,c,binomial(r+c-2,c-1)))); for(i=0,m,for(r=0,m,R[r+1]=Vec(sum(c=0,m,x^c*Ser(R[c+1])^r+O(x^(m+1))))));R[7][n+1]}

Formula

G.f.: A(x) = Sum_{n>=0} x^n*R_n(x)^6, where R_n(x) is the g.f. of row n in table A124460 and satisfies: R_n(x) = Sum_{k>=0} x^k * R_k(x)^n for n>=0.

A124467 Main diagonal of rectangular table A124460.

Original entry on oeis.org

1, 2, 8, 44, 302, 2437, 22143, 220663, 2377322, 27462836, 338270681, 4422338285, 61114545578, 889542935517, 13593252066134, 217461100417945, 3632859088845390, 63233532604594586, 1144455373221119025
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Comments

In table A124460, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.

Crossrefs

Cf. A124460 (table); rows: A124461, A124462, A124463, A124464, A124465, A124466; A124468 (diagonal).

Programs

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

A124468 Secondary diagonal of rectangular table A124460.

Original entry on oeis.org

1, 3, 13, 73, 506, 4117, 37703, 378529, 4106995, 47766886, 592228264, 7791350900, 108323265571, 1585742194947, 24364012940723, 391780319558565, 6577005078357973, 115010285695135090, 2090723838361780991
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Comments

In table A124460, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.

Crossrefs

Cf. A124460 (table); rows: A124461, A124462, A124463, A124464, A124465, A124466; A124467 (diagonal).

Programs

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

A124469 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124460.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 22, 28, 11, 1, 1, 65, 120, 81, 20, 1, 1, 209, 500, 494, 219, 37, 1, 1, 730, 2088, 2733, 1812, 578, 70, 1, 1, 2743, 8884, 14411, 12904, 6299, 1518, 135, 1, 1, 10958, 38803, 74484, 84424, 56590, 21384, 4007, 264, 1, 1, 46057, 174366
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Comments

In table A124460, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.

Examples

			Triangle begins:
1;
1, 1;
1, 3, 1;
1, 8, 6, 1;
1, 22, 28, 11, 1;
1, 65, 120, 81, 20, 1;
1, 209, 500, 494, 219, 37, 1;
1, 730, 2088, 2733, 1812, 578, 70, 1;
1, 2743, 8884, 14411, 12904, 6299, 1518, 135, 1;
1, 10958, 38803, 74484, 84424, 56590, 21384, 4007, 264, 1;
1, 46057, 174366, 383391, 526121, 453082, 238853, 72076, 10693, 521, 1;

Crossrefs

Cf. A124470 (row sums), A006127 (diagonal T(n+1, n)); A124460 (table).

Programs

  • PARI
    {T(n,k)=local(R=vector(n+2,r,vector(n+2,c,binomial(r+c-2,c-1)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^r+O(x^(n+1)))))); Vec(subst(Ser(vector(n+1,j,R[j][n+1])),x,x/(1+x))/(1+x))[k+1]}

Formula

Secondary diagonal T(n+1,n) = 2^n + n = A006127(n).

A124470 Row sums of the triangle A124469, in which row n equals the inverse binomial transform of column n in the triangle A124460.

Original entry on oeis.org

1, 2, 5, 16, 63, 288, 1461, 8013, 46896, 290916, 1905162, 13131193, 95005565, 719862361, 5700083435, 47075439047, 404760508374, 3617018777976, 33539368302476, 322222320833572, 3202873385842352, 32895350423011349
Offset: 0

Views

Author

Paul D. Hanna, Nov 03 2006

Keywords

Comments

In table A124460, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.

Crossrefs

Cf. A124469, A124460.

Programs

  • PARI
    {a(n)=local(R=vector(n+2,r,vector(n+2,c,binomial(r+c-2,c-1)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^r+O(x^(n+1)))))); subst(truncate(subst(Ser(vector(n+1,j,R[j][n+1])),x,x/(1+x))/(1+x)),x,1)}
Showing 1-10 of 14 results. Next

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

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