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 1-8 of 8 results.

A161797 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3).

Original entry on oeis.org

1, 1, 4, 16, 71, 336, 1660, 8464, 44207, 235306, 1271807, 6961307, 38508659, 214950425, 1209170536, 6848080767, 39014400171, 223439516338, 1285660965508, 7428738358924, 43087099589998, 250766507928988, 1464026402082801
Offset: 0

Views

Author

Paul D. Hanna, Jun 19 2009

Keywords

Links

Crossrefs

Cf. A109081, A321798, A321799.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]/(n-k+1) * Binomial[n+2*k-1,n-k], {k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Nov 18 2017 *)
  • PARI
    {a(n,m=1)=sum(k=0,n,binomial(n+m-1,k)*m/(n-k+m)*binomial(n+2*k-1,n-k))}

Formula

a(n) = Sum_{k=0..n} C(n,k)/(n-k+1) * C(n+2*k-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
a(n,m) = Sum_{k=0..n} C(n+m-1,k)*m/(n-k+m) * C(n+2*k-1,n-k).
G.f.: A(x) = (1/x)*serreverse[x/(1 + x/(1 - x)^3)].
Recurrence: 3*(n+1)*(3*n - 2)*(3*n + 2)*(2145*n^4 - 14355*n^3 + 33844*n^2 - 32668*n + 10380)*a(n) = 3*(115830*n^7 - 833085*n^6 + 2195691*n^5 - 2521863*n^4 + 998671*n^3 + 259048*n^2 - 263292*n + 41520)*a(n-1) + 3*(n-2)*(19305*n^6 - 129195*n^5 + 315651*n^4 - 367201*n^3 + 219176*n^2 - 66584*n + 7608)*a(n-2) + 3*(n-3)*(n-2)*(64350*n^5 - 334125*n^4 + 546005*n^3 - 255608*n^2 - 71320*n + 54328)*a(n-3) - 23*(n-4)*(n-3)*(n-2)*(2145*n^4 - 5775*n^3 + 3649*n^2 + 535*n - 654)*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt(sqrt((159 + 100*sqrt(3))/13) - 2 - 5/sqrt(3)) * (3 + 2*sqrt(3) + sqrt(153 + 100*sqrt(3))/3)^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 5/2)). - Vaclav Kotesovec, Nov 18 2017

A365114 G.f. satisfies A(x) = 1 + x / (1 - x*A(x))^4.

Original entry on oeis.org

1, 1, 4, 14, 56, 241, 1080, 4998, 23704, 114588, 562552, 2797138, 14057140, 71288385, 364360204, 1874960408, 9706035408, 50510552881, 264096980192, 1386676113360, 7308650513232, 38654087828310, 205076534841112, 1091144400876394, 5820924498941668
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2023

Keywords

Crossrefs

Cf. A000108, A365113, A365115.
Cf. A321798, A365111.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x/(1 - x*A(x))^s, then a(n) = Sum_{k=0..n} binomial(n-k+1,k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).

A321799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^5).

Original entry on oeis.org

1, 1, 6, 31, 176, 1071, 6797, 44493, 298279, 2037550, 14131441, 99244564, 704360703, 5043969503, 36399930179, 264451303466, 1932650461883, 14198082537190, 104792195449688, 776681663951998, 5778226417888171, 43135097969972931, 323012620411650708, 2425745980876575899, 18264470545275495152
Offset: 0

Views

Author

Ludovic Schwob, Nov 19 2018

Keywords

Links

Crossrefs

Cf. A109081, A161797, A321798.

Programs

  • GAP
    List([0..25],n->Sum([0..n],k->Binomial(n,k)/(n-k+1)*Binomial(n+4*k-1,n-k))); # Muniru A Asiru, Nov 24 2018
  • Magma
    [1] cat [&+[(Binomial(n,k)/(n-k+1)) * Binomial(n+4*k-1,n-k):  k in [0..n]]: n in [1.. 25]]; // Vincenzo Librandi, Dec 08 2018
  • Maple
    eq:= a - 1/(1-x/(1-x*a)^5):
S:= series(RootOf(numer(eq),a),x,31):
seq(coeff(S,x,j),j=0..30); # Robert Israel, Dec 10 2018
  • Mathematica
    a[n_]:=Sum[ Binomial[n,k]/(n-k+1)*Binomial[n+4*k-1,n-k], {k,0,n}]; Array[a, 20, 0] (* Stefano Spezia, Nov 19 2018 *)
A[] = 0; Do[A[x] = 1/(1-x/(1-x*A[x])^5)+O[x]^25, {25}];
CoefficientList[A[x], x] (* Jean-François Alcover, Dec 30 2018 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(n+4*k-1, n-k)/(n-k+1)); \\ Michel Marcus, Nov 19 2018
  • Sage
    [sum(binomial(n,k)*binomial(n+4*k-1,n-k)/(n-k+1) for k in (0..n)) for n in range(25)] # G. C. Greubel, Dec 14 2018

Formula

a(n) = Sum_{k=0..n} (C(n,k)/(n-k+1)) * C(n+4*k-1,n-k).
a(n) ~ sqrt((1 - r*s)*(1 + 4*r*s) / (5*Pi*(5*s - 2))) / (2 * n^(3/2) * r^(n+1)), where r = 0.124910212976238209867004924637837518925706044646... and s = 1.72708330560542094133450070142549940430523638921... are real roots of the system of equations s*(1 - r/(1 - r*s)^5) = 1, 5*r^2*s^2 = (1 - r*s)^6. - Vaclav Kotesovec, Nov 21 2018

A365087 G.f. satisfies A(x) = 1 + x*A(x) / (1 + x*A(x))^4.

Original entry on oeis.org

1, 1, -3, -1, 29, -44, -265, 1114, 1369, -19076, 20388, 250977, -875281, -2116594, 19136754, -7765108, -306092007, 830209808, 3388957208, -22266676364, -8185922076, 413223401045, -814031607979, -5513566634947, 27558060911119, 35395095404776
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2023

Keywords

Crossrefs

Cf. A090192, A365085, A365086, A365088.
Cf. A321798, A364737, A365083.

Programs

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

Formula

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

A367235 G.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - x*A(x))^4.

Original entry on oeis.org

1, 1, 7, 50, 399, 3422, 30798, 286974, 2744947, 26798010, 265945022, 2674970684, 27209385886, 279412999031, 2892787737002, 30161921520976, 316440334960563, 3338105334701396, 35385133077851602, 376732207920371784, 4026682585718602014
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Links

Crossrefs

Cf. A321798, A365114, A367234.
Cf. A108447, A367232, A367233.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).

A367234 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x))^4.

Original entry on oeis.org

1, 1, 6, 35, 226, 1561, 11276, 84150, 643730, 5021038, 39781858, 319282210, 2590312872, 21208628405, 175024439504, 1454329099044, 12157356271998, 102170610282040, 862721635191860, 7315768816166027, 62274763166575410, 531950072655682896, 4558282056420235664
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Links

Crossrefs

Cf. A321798, A365114, A367235.
Cf. A001003, A011270, A365150.
Cf. A361306, A378567.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).
From Seiichi Manyama, Dec 01 2024: (Start)
G.f.: exp( Sum_{k>=1} A378567(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x)^4)^(n+1).
G.f.: (1/x) * Series_Reversion( x*(1 - x/(1 - x)^4) ). (End)

A349023 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4)^2.

Original entry on oeis.org

1, 2, 11, 64, 417, 2892, 20941, 156500, 1198049, 9346690, 74042938, 594001236, 4815995027, 39399831458, 324840184326, 2696343599336, 22514057175337, 188977375146888, 1593661234493561, 13495942411592260, 114723671513478118, 978570384358686064
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2021

Keywords

Crossrefs

Cf. A118971, A321798, A349024.

Programs

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

Formula

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).

A349024 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4)^3.

Original entry on oeis.org

1, 3, 18, 124, 951, 7764, 66200, 582594, 5252133, 48254668, 450186720, 4253328540, 40612877001, 391300954065, 3799506069816, 37142836241690, 365255937037437, 3610755090793272, 35861607622930556, 357670540310182842, 3580797575489620740
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2021

Keywords

Crossrefs

Cf. A118971, A321798, A349023.

Programs

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

Formula

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
Showing 1-8 of 8 results.

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