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.

Previous Showing 11-17 of 17 results.

A322248 G.f.: 1/sqrt( (1 + 3*x)*(1 - 13*x) ).

Original entry on oeis.org

1, 5, 57, 605, 6961, 81525, 973545, 11765325, 143522145, 1763351525, 21789466777, 270509191485, 3371353189009, 42155188480085, 528587607974217, 6644129071092525, 83691484792766145, 1056178325362832325, 13351036742005533945, 169019946403985898525, 2142600388730167543281, 27193744661180635582005, 345520219114720175821737, 4394534009569783690837005, 55943630366450131877449761, 712778930909503993783945125
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2018

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 57*x^2 + 605*x^3 + 6961*x^4 + 81525*x^5 + 973545*x^6 + 11765325*x^7 + 143522145*x^8 + 1763351525*x^9 + 21789466777*x^10 + ...
such that A(x)^2 = 1/(1 - 10*x - 39*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 5*x + 41*x^2 + 365*x^3 + 3537*x^4 + 35925*x^5 + 378105*x^6 + 4084925*x^7 + 45044129*x^8 + 504880805*x^9 + 5735247817*x^10 + ...

Links

Crossrefs

Cf. A084605, A098409, A248168.
Cf. A322249.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[((1+3x)(1-13x))],{x,0,30}],x] (* Harvey P. Dale, Jun 29 2021 *)
  • PARI
    /* Using generating function: */
{a(n) = polcoeff( 1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Using binomial formula: */
{a(n) = sum(k=0,n, (-3)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Using binomial formula: */
{a(n) = sum(k=0,n, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* a(n) as a central coefficient */
{a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
a(n) equals the (central) coefficient of x^n in (1 + 5*x + 16*x^2)^n.
a(n) ~ 13^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 10 2018
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) + 39*(n-1)*a(n-2) for n > 1. - Seiichi Manyama, Apr 22 2019
a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A385813 Expansion of 1/((1-3*x) * (1-7*x))^(3/2).

Original entry on oeis.org

1, 15, 156, 1400, 11655, 92925, 721140, 5496300, 41361255, 308344025, 2282167272, 16795140180, 123030071437, 897791417775, 6530377362480, 47370038320800, 342794475282915, 2475479922896925, 17843821672113780, 128412824128709400, 922775179449162501, 6622378039719342615
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A385563, A385728.
Cf. A098409, A182401.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 25 2025
  • Mathematica
    CoefficientList[Series[1/((1-3x)*(1-7*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 25 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(3/2))

Formula

n*a(n) = (10*n+5)*a(n-1) - 21*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A182401(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (5/2)^k * (-21/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).

A387278 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 10, 78, 560, 3885, 26550, 180285, 1221400, 8272251, 56062550, 380361212, 2583867720, 17575724491, 119705522370, 816297170310, 5572945684800, 38088275031435, 260576833989150, 1784382167211378, 12229792774162800, 83888652677196591, 575858959975595010
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2025

Keywords

Links

Crossrefs

Cf. A098409, A387275, A387276, A387277.
Cf. A026376, A344055.

Programs

  • Magma
    [&+[3^(n-k) * Binomial(n+1,k+1) * Binomial(2*k+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 30 2025
  • Mathematica
    Table[Sum[3^(n-k)*Binomial[n+1,k+1]*Binomial[2*k+2,k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 30 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^(n-k)*binomial(n+1, k+1)*binomial(2*k+2, k+2));

Formula

n*(n+2)*a(n) = (n+1) * (5*(2*n+1)*a(n-1) - 21*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+5*x+x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 2*x), with offset 1.

A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

Original entry on oeis.org

1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2019

Keywords

Comments

Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.

Links

Crossrefs

Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))
  • PARI
    {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}

Formula

a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 30 2019

A383948 Expansion of 1/sqrt((1-3*x)^3 * (1-7*x)).

Original entry on oeis.org

1, 8, 51, 308, 1855, 11340, 70665, 448320, 2887155, 18815240, 123759097, 819969276, 5464090177, 36580917716, 245837438055, 1657396783440, 11204207037315, 75918595916520, 515462211835305, 3506072423912940, 23885410548196701, 162951783575205108, 1113110415733083531
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A098409, A360318.
Cf. A383949, A383950.
Cf. A132894.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 3*x)^3 * (1-7*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    CoefficientList[Series[ 1/Sqrt[(1-3*x)^3*(1-7*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-3*x)^3*(1-7*x)))

Formula

n*a(n) = (10*n-2)*a(n-1) - 21*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).

A117852 Mirror image of A098473 formatted as a triangular array.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 20, 18, 6, 1, 70, 80, 36, 8, 1, 252, 350, 200, 60, 10, 1, 924, 1512, 1050, 400, 90, 12, 1, 3432, 6468, 5292, 2450, 700, 126, 14, 1, 12870, 27456, 25872, 14112, 4900, 1120, 168, 16, 1, 48620, 115830, 123552, 77616, 31752, 8820, 1680, 216, 18, 1
Offset: 0

Views

Author

Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006

Keywords

Examples

			Triangle begins:
    1;
    2,   1;
    6,   4,   1;
   20,  18,   6,   1;
   70,  80,  36,   8,   1;
  252, 350, 200,  60,  10,   1;
  ...

Links

Crossrefs

Cf. A098473.

Programs

  • Maple
    c:=n->binomial(2*n, n): T:=proc(n, k) if k<=n then binomial(n, k)*c(n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; #
  • Mathematica
    Table[ Binomial[n, k]*Binomial[2*n - 2*k, n - k], {n,0,10}, {k,0,n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

Formula

Sum_{k=0..n} T(n,k)*x^k = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007
T(n,k) = binomial(n,k)*A000984(n-k). - Philippe Deléham, Dec 12 2009
O.g.f.: 1/sqrt( (1 - x*t)*(1 - (x + 4)*t) ) = 1 + (2 + x)*t + (6 + 4*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 12 2007

A293491 a(n) = n! * [x^n] exp((n+2)*x)*BesselI(0,2*x).

Original entry on oeis.org

1, 3, 18, 155, 1734, 23877, 390804, 7417377, 160256070, 3885021569, 104465601756, 3086353547433, 99399100528924, 3466411543407555, 130151205663179112, 5235127829223881895, 224609180728848273990, 10239557195235638377449, 494317596005491398892620, 25192788307121307053168673
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 10 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A000984.

Links

Crossrefs

Cf. A000984, A026375, A081671, A098409, A098410, A104454, A186925, A292627, A292632.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 x], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/Sqrt[(1 - n x) (1 - (n + 4) x)], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[Binomial[n, k] Binomial[2 k, k] n^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[(n + 2)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4/(2 + n)^2], {n, 0, 19}]

Formula

a(n) = [x^n] 1/sqrt((1 - n*x)*(1 - (n + 4)*x)).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*n^(n-k).
a(n) ~ exp(2) * BesselI(0,2) * n^n. - Vaclav Kotesovec, Oct 16 2017
Previous Showing 11-17 of 17 results.

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

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