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 12 results. Next

A346626 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - x).

Original entry on oeis.org

1, 2, 8, 44, 280, 1936, 14128, 107088, 834912, 6652608, 53934080, 443467136, 3689334272, 30997608960, 262651640064, 2241857334528, 19257951946240, 166362924583936, 1444351689281536, 12595885932259328, 110287974501355520, 969178569410404352, 8544982917273509888, 75565732555028701184
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2021

Keywords

Comments

Partial sums of A213282.

Links

Crossrefs

Cf. A001764, A006318, A199475, A213282, A349310, A349311, A349312, A349313, A349314.

Programs

  • Mathematica
    nmax = 23; A[] = 0; Do[A[x] = (1 + x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 23; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/(1 - x)^(3 k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 23}]

Formula

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 - x)^(3*k+1).
a(0) = 1; a(n) = a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ 2^(n - 1/2) / (sqrt(3*Pi*(2 - (2 - sqrt(2))^(1/3)/2^(2/3) - 1/(2*(2 - sqrt(2)))^(1/3))) * n^(3/2) * (2 - 3/(sqrt(2) - 1)^(1/3) + 3*(sqrt(2) - 1)^(1/3))^n). - Vaclav Kotesovec, Nov 04 2021
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-k) * binomial(n,k) * binomial(2*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A213336 G.f. satisfies A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 8, 64, 568, 5440, 54888, 574848, 6190872, 68132224, 762874568, 8663106496, 99536424952, 1155012037824, 13516570396968, 159340702404352, 1890451582396632, 22555522916988672, 270466907608087944, 3257754635421506368, 39397587357527547320
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 64*x^3 + 568*x^4 + 5440*x^5 + 54888*x^6 +...
G.f.: A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Links

Crossrefs

Cf. A213335, A002293; variants: A006319, A213282.
Partial sums give A349310. - Seiichi Manyama, Oct 03 2023

Programs

  • PARI
    /* G.f. A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4: */
{a(n)=local(A, G=1+x); for(i=1, n, G=1+x*G^4+x*O(x^n)); A=subst(G, x, x/(1-x+x*O(x^n))^4); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* G.f. A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4: */
{a(n)=local(F=1+x+x*O(x^n),A=1); for(i=1, n+1, F=1+x/subst(F^4, x, -x+x*O(x^n))); A=(serreverse(x/F^4)/x)^(1/4);polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4 is the g.f. of A213335.
G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) = Sum_{k=0..n} binomial(n+3*k-1,n-k) * binomial(4*k,k)/(3*k+1). - Seiichi Manyama, Oct 03 2023

A213281 G.f. satisfies: A(x) = 1 + x/A(-x)^3.

Original entry on oeis.org

1, 1, 3, -3, -35, 48, 693, -1046, -16635, 26328, 442396, -720327, -12541509, 20810208, 371430414, -624691212, -11356013899, 19293440712, 355703260500, -609103135196, -11355804637164, 19568456886336, 368147199241021, -637674031240302, -12087185276792061
Offset: 0

Views

Author

Paul D. Hanna, Jun 08 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 - 3*x^3 - 35*x^4 + 48*x^5 + 693*x^6 - 1046*x^7 +...
where
1/A(-x) = 1 + x - 2*x^2 - 8*x^3 + 30*x^4 + 143*x^5 - 638*x^6 - 3272*x^7 +...
x/A(-x)^3 = x + 3*x^2 - 3*x^3 - 35*x^4 + 48*x^5 + 693*x^6 - 1046*x^7 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 10*x^3 - 87*x^4 - 102*x^5 + 1632*x^6 + 1974*x^7 +...
The g.f. G(x) of A213282 begins:
G(x) = 1 + x + 6*x^2 + 36*x^3 + 236*x^4 + 1656*x^5 + 12192*x^6 + 92960*x^7 +...
where G(x) = A(x*G(x)^3) and G(x/A(x)^3) = A(x);
also, G(x) = F(x/(1-x)^3) where F(x) = 1 + x*F(x)^3 is g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

Links

Crossrefs

Cf. A213282, A213252, A213335; A001764.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x/subst(A^3,x,-x+x*O(x^n)));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = G(x/A(x)^3) where G(x) = A(x*G(x)^3) is the g.f. of A213282.
G.f. satisfies: A(x) = ( x/Series_Reversion( x*F(x/(1-x)^3)^3 ) )^(1/3) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
G.f. satisfies: A(x) = A(x)*A(-x) + x/A(x)^2.

A366431 G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^(5/2).

Original entry on oeis.org

1, 1, 5, 25, 135, 775, 4651, 28845, 183450, 1190050, 7844230, 52389678, 353770190, 2411324700, 16568343325, 114639216915, 798076174113, 5586035989185, 39287407321075, 277508001643575, 1967816928168265, 14003018984540741, 99965175670335750
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366400.
Cf. A006319, A071724, A213282, A213336, A366432, A366433, A366434, A366435, A366436, A366437.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k/2-1, n-k)*binomial(5*k/2, k)/(3*k/2+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k/2-1,n-k) * binomial(5*k/2,k) / (3*k/2+1).

A366499 G.f. A(x) satisfies A(x) = 1 + x / ((1+x)^3*A(x)^2).

Original entry on oeis.org

1, 1, -5, 25, -145, 945, -6641, 49057, -375361, 2948353, -23634049, 192554753, -1589812225, 13272519937, -111850866433, 950220134913, -8129133081601, 69971682467841, -605546841831425, 5265763716550657, -45988028107350017, 403192288488677377
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2023

Keywords

Crossrefs

Cf. A001006, A366221, A366272, A366273, A366495, A366496, A366497, A366498, A366500, A366501.
Cf. A213282.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(n+2*k-1, n-k)*binomial(3*k-1, k)/(3*k-1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A213282.
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(n+2*k-1,n-k) * binomial(3*k-1,k) / (3*k-1).

A366176 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3/(1 - x)^2.

Original entry on oeis.org

1, 1, 5, 27, 161, 1030, 6921, 48190, 344669, 2517303, 18695908, 140771477, 1072130229, 8244820518, 63931532190, 499308229278, 3924204043333, 31012883225891, 246304580923299, 1964794017165157, 15735626383151876, 126476316316459089, 1019883740031357941
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2023

Keywords

Crossrefs

Partial sums give A199475.
Cf. A001764, A213282, A307678.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, n-k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+k-1,n-k) * binomial(3*k,k)/(2*k+1).

A361932 G.f. A(x) satisfies A(x) = 1 + (x * A(x) / (1 - x))^3.

Original entry on oeis.org

1, 0, 0, 1, 3, 6, 13, 33, 84, 208, 522, 1341, 3476, 9042, 23673, 62426, 165504, 440664, 1178168, 3162357, 8517681, 23013294, 62356329, 169408107, 461366499, 1259311824, 3444497550, 9439766700, 25916832981, 71274793968, 196325540206, 541579442133
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2023

Keywords

Crossrefs

Partial sums give A071879.
Cf. A000245, A213282, A364403.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-1, n-3*k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k) * binomial(3*k,k) / (2*k+1).

A364403 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^3.

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 6, 10, 18, 39, 91, 204, 435, 919, 1992, 4434, 9947, 22215, 49455, 110480, 248505, 561930, 1273610, 2889666, 6566736, 14959083, 34163511, 78182700, 179201199, 411325125, 945512784, 2176710450, 5018195400, 11583688995, 26770164919
Offset: 0

Views

Author

Seiichi Manyama, Oct 15 2023

Keywords

Crossrefs

Partial sums give A364552.
Cf. A000245, A213282, A361932.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-k-1, n-4*k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-k-1,n-4*k) * binomial(3*k,k) / (2*k+1).

A379254 G.f. A(x) satisfies A(x) = ( (1 + x*A(x))/(1 - x*A(x)^2) )^3.

Original entry on oeis.org

1, 6, 72, 1136, 20496, 400176, 8230592, 175643712, 3852905472, 86338960640, 1967950718976, 45483461999616, 1063433350498304, 25107661327202304, 597766180153565184, 14335020154675867648, 345948883288769740800, 8395511682729703931904
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A213282, A365843, A379255.
Cf. A379256.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+4*k+2, k)*binomial(3*n+3*k+3, n-k)/(n+k+1));

Formula

G.f.: B(x)^3 where B(x) is the g.f. of A379256.
a(n) = Sum_{k=0..n} binomial(3*n+4*k+2,k) * binomial(3*n+3*k+3,n-k)/(n+k+1).

A379255 G.f. A(x) satisfies A(x) = ( (1 + x*A(x))/(1 - x*A(x)^3) )^3.

Original entry on oeis.org

1, 6, 90, 1910, 47250, 1274406, 36344906, 1077809718, 32899427106, 1026823733702, 32619190553274, 1051205539768566, 34282637873690290, 1129326395659189734, 37522172645425790634, 1255954522404101871286, 42312438228338307500610, 1433621819994034883749254
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A213282, A365843, A379254.
Cf. A379257.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+7*k+2, k)*binomial(3*n+6*k+3, n-k)/(n+2*k+1));

Formula

G.f.: B(x)^3 where B(x) is the g.f. of A379257.
a(n) = Sum_{k=0..n} binomial(3*n+7*k+2,k) * binomial(3*n+6*k+3,n-k)/(n+2*k+1).
Showing 1-10 of 12 results. Next

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