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 1-6 of 6 results.

A385716 Expansion of 1/((1-x) * (1-13*x))^(3/2).

Original entry on oeis.org

1, 21, 348, 5320, 78135, 1120287, 15805972, 220445316, 3047961735, 41857891075, 571725145992, 7774356136092, 105324231178621, 1422411298153125, 19157947746089520, 257427540725705056, 3451990965984505251, 46205867184493459023, 617482101788090727220, 8239952016851603641320
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A331516, A385563.
Cf. A340973, A386362.

Programs

  • Mathematica
    Module[{x}, CoefficientList[Series[1/((1-x)*(1-13*x))^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-13*x))^(3/2))

Formula

n*a(n) = (14*n+7)*a(n-1) - 13*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^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^k * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A386362(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (7/2)^k * (-13/14)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(3*n) * 13^(n + 3/2) / (36*sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

A385563 Expansion of 1/((1-x) * (1-5*x))^(3/2).

Original entry on oeis.org

1, 9, 60, 360, 2055, 11403, 62132, 334260, 1781415, 9425295, 49581576, 259601004, 1353939405, 7038232425, 36484340400, 188665670880, 973545780195, 5014258620075, 25783103206100, 132378800689800, 678768332410245, 3476164133573505, 17782899991147500
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Partial sums of A383254.
Cf. A331516, A385716.
Cf. A385728, A385813.
Cf. A002212, A383949.

Programs

  • Mathematica
    Module[{a, n}, RecurrenceTable[{a[n] == ((6*n+3)*a[n-1] - 5*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 9}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(3/2))

Formula

n*a(n) = (6*n+3)*a(n-1) - 5*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A002212(n+1).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (3/2)^k * (-5/6)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(n) * 5^(n + 3/2) / (4*sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

A331514 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(1 - 2*k*x + ((k-2)*x)^2)^(3/2).

Original entry on oeis.org

1, 1, 0, 1, 3, -6, 1, 6, 6, 0, 1, 9, 30, 10, 30, 1, 12, 66, 140, 15, 0, 1, 15, 114, 450, 630, 21, -140, 1, 18, 174, 1000, 2955, 2772, 28, 0, 1, 21, 246, 1850, 8430, 18963, 12012, 36, 630, 1, 24, 330, 3060, 18855, 69384, 119812, 51480, 45, 0
Offset: 0

Views

Author

Seiichi Manyama, Jan 19 2020

Keywords

Examples

			Square array begins:
    1,  1,    1,     1,     1,      1, ...
    0,  3,    6,     9,    12,     15, ...
   -6,  6,   30,    66,   114,    174, ...
    0, 10,  140,   450,  1000,   1850, ...
   30, 15,  630,  2955,  8430,  18855, ...
    0, 21, 2772, 18963, 69384, 187425, ...

Links

Crossrefs

Columns k=1..5 give A000217(n+1), A002457, A002695(n+1), A331515, A331516.
Cf. A307883, A331511.

Programs

  • Mathematica
    T[n_, k_] = 1/2 * Sum[If[k == 2 && n == j - 1, 1, (k - 2)^(n + 1 - j)] * j * Binomial[n + 1, j] * Binomial[n + 1 + j, j], {j, 1, n + 1}]; Table[Table[T[n, k - n], {n, 0, k}], {k, 0, 9}] //Flatten (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    T(n,k) = (1/2)*sum(j=1,n+1,(k-2)^(n+1-j)*j*binomial(n+1,j)*binomial(n+1+j,j));
matrix(7, 7, n, k, T(n-1, k-1)) \\ Michel Marcus, Jan 20 2020

Formula

T(n,k) = (1/2) * Sum_{j=1..n+1} (k-2)^(n+1-j) * j * binomial(n+1,j) * binomial(n+1+j,j).
n * T(n,k) = k * (2*n+1) * T(n-1,k) - (k-2)^2 * (n+1) * T(n-2,k) for n > 1.
T(n,k) = ((n+2)/2) * Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
T(n,k) = Sum_{j=0..n} (k/2)^j * (-(k-2)^2/(2*k))^(n-j) * (2*j+1) * binomial(2*j,j) * binomial(j,n-j) for k > 0. - Seiichi Manyama, Aug 20 2025

A331793 Expansion of ((1 - 5*x)/sqrt(1 - 10*x + 9*x^2) - 1)/(8*x^2).

Original entry on oeis.org

1, 10, 87, 740, 6285, 53550, 458115, 3934600, 33913881, 293244050, 2542684463, 22101612780, 192530903461, 1680415209270, 14692052109915, 128653303453200, 1128147127156785, 9905115333850650, 87066787614156807, 766127762539955700, 6747880819438628541
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2020

Keywords

Links

Crossrefs

Column 5 of A331791.
Cf. A331516.

Programs

  • Mathematica
    a[n_] := Sum[4^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, May 05 2021 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(((1-5*x)/sqrt(1-10*x+9*x^2)-1)/(8*x^2))
  • PARI
    a(n) = sum(k=0, n, 4^k*binomial(n+1, k)*binomial(n+1, k+1));

Formula

a(n) = (2/(n+2)) * A331516(n) = Sum_{k=0..n} 4^k * binomial(n+1,k) * binomial(n+1,k+1).
n * (n+2) * a(n) = (n+1) * (5 * (2*n+1) * a(n-1) - 9 * n * a(n-2)) for n>1.
a(n) ~ 3^(2*n + 3) / (2^(5/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 26 2020
From Seiichi Manyama, Aug 23 2025: (Start)
a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+1,k+1) * binomial(2*k+2,k+2). (End)
From Seiichi Manyama, Aug 25 2025: (Start)
a(n) = [x^n] (1+5*x+4*x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 4*x) / 2, with offset 1. (End)

A383946 Expansion of 1/sqrt((1-9*x)^3 * (1-x)).

Original entry on oeis.org

1, 14, 159, 1676, 17005, 168570, 1645035, 15873240, 151863705, 1443272870, 13643264503, 128404376292, 1204055841157, 11255397745298, 104933302809795, 976016662472880, 9059771065058865, 83945271527170110, 776569280469986895, 7173673630527966780, 66182347507155379101, 609866573826736447914
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2025

Keywords

Links

Crossrefs

Cf. A084771, A331516.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 9*x)^3 * (1-x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    CoefficientList[Series[1/Sqrt[(1-9*x)^3*(1-x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-9*x)^3*(1-x)))

Formula

a(n) = A331516(n) - A331516(n-1).
n*a(n) = (10*n+4)*a(n-1) - 9*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).

A387313 Expansion of 1/((1-x) * (1-9*x))^(5/2).

Original entry on oeis.org

1, 25, 415, 5775, 72870, 864150, 9818130, 108109650, 1162302735, 12262882775, 127424209913, 1307536637225, 13276264807260, 133597932407100, 1334029357684980, 13231465264538100, 130461712570627245, 1279632533997010725, 12492837802976030115, 121456026730456739475
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A084771, A331516, A387314.
Cf. A387229, A387307.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-9*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-9*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-9*x))^(5/2))

Formula

n*a(n) = (10*n+15)*a(n-1) - 9*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-8)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 8^k * 9^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * A387307(n).
a(n) = (-1)^n * Sum_{k=0..n} 10^k * (9/10)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.