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 1-4 of 4 results.

A292627 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*BesselI(0,2*x).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 2, 3, 0, 1, 3, 6, 7, 6, 1, 4, 11, 20, 19, 0, 1, 5, 18, 45, 70, 51, 20, 1, 6, 27, 88, 195, 252, 141, 0, 1, 7, 38, 155, 454, 873, 924, 393, 70, 1, 8, 51, 252, 931, 2424, 3989, 3432, 1107, 0, 1, 9, 66, 385, 1734, 5775, 13236, 18483, 12870, 3139, 252, 1, 10, 83, 560, 2995, 12276, 36645, 73392, 86515, 48620, 8953, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

Comments

A(n,k) is the k-th binomial transform of A126869 evaluated at n.

Examples

			E.g.f. of column k: A_k(x) =  1 + k*x/1! + (k^2 + 2)*x^2/2! + (k^3 + 6*k)*x^3/3! + (k^4 + 12*k^2 + 6)*x^4/4! + (k^5 + 20*k^3 + 30*k)*x^5/5! + ...
Square array begins:
  1,   1,    1,    1,     1,     1,  ...
  0,   1,    2,    3,     4,     5,  ...
  2,   3,    6,   11,    18,    27,  ...
  0,   7,   20,   45,    88,   155,  ...
  6,  19,   70,  195,   454,   931,  ...
  0,  51,  252,  873,  2424,  5775,  ...

Links

Crossrefs

Columns k=0..7 give A126869, A002426, A000984, A026375, A081671, A098409, A098410, A104454.
Rows n=0..2 give A000012, A001477, A059100.
Main diagonal gives A186925.
Cf. A292628, A307847.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[0, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/Sqrt[(1 + 2 x - k x) (1 - 2 x - k x)], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

O.g.f. of column k: 1/sqrt( (1 - (k-2)*x)*(1 - (k+2)*x) ).
E.g.f. of column k: exp(k*x)*BesselI(0,2*x).
From Seiichi Manyama, May 01 2019: (Start)
A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + x^2)^n.
A(n,k) = Sum_{j=0..n} (k-2)^(n-j) * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (k+2)^(n-j) * (-1)^j * binomial(n,j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - (k^2-4) * (n-1) * A(n-2,k). (End)
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-2*j) * binomial(n,2*j) * binomial(2*j,j). - Seiichi Manyama, May 04 2019
T(n,k) = (1/Pi) * Integral_{x = -1..1} (k - 2 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (k + 2 - 4*x^2)^n/sqrt(1 - x^2) dx. - Peter Bala, Jan 27 2020
A(n,k) = (1/4)^n * Sum_{j=0..n} (k-2)^j * (k+2)^(n-j) * binomial(2*j,j) * binomial(2*(n-j),n-j). - Seiichi Manyama, Aug 18 2025

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 13, 19, 1, 1, 1, 9, 19, 49, 51, 1, 1, 1, 11, 25, 91, 161, 141, 1, 1, 1, 13, 31, 145, 331, 581, 393, 1, 1, 1, 15, 37, 211, 561, 1441, 2045, 1107, 1, 1, 1, 17, 43, 289, 851, 2841, 5797, 7393, 3139, 1
Offset: 0

Views

Author

Seiichi Manyama, May 01 2019

Keywords

Examples

			Square array begins:
   1,   1,    1,    1,     1,     1,     1, ...
   1,   1,    1,    1,     1,     1,     1, ...
   1,   3,    5,    7,     9,    11,    13, ...
   1,   7,   13,   19,    25,    31,    37, ...
   1,  19,   49,   91,   145,   211,   289, ...
   1,  51,  161,  331,   561,   851,  1201, ...
   1, 141,  581, 1441,  2841,  4901,  7741, ...
   1, 393, 2045, 5797, 12489, 22961, 38053, ...

Links

Crossrefs

Columns k=0..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
Main diagonal gives A187018.
Cf. A110180, A292627, A307847, A307860, A307910.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

Formula

A(n,k) is the coefficient of x^n in the expansion of (1 + x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * binomial(2*j,j).
D-finite with recurrence: n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

A307910 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*k*x + k*(k-4)*x^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 7, 0, 1, 4, 15, 32, 19, 0, 1, 5, 24, 81, 136, 51, 0, 1, 6, 35, 160, 459, 592, 141, 0, 1, 7, 48, 275, 1120, 2673, 2624, 393, 0, 1, 8, 63, 432, 2275, 8064, 15849, 11776, 1107, 0, 1, 9, 80, 637, 4104, 19375, 59136, 95175, 53344, 3139, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2019

Keywords

Examples

			Square array begins:
   1,   1,     1,     1,      1,       1,       1, ...
   0,   1,     2,     3,      4,       5,       6, ...
   0,   3,     8,    15,     24,      35,      48, ...
   0,   7,    32,    81,    160,     275,     432, ...
   0,  19,   136,   459,   1120,    2275,    4104, ...
   0,  51,   592,  2673,   8064,   19375,   40176, ...
   0, 141,  2624, 15849,  59136,  168125,  400896, ...
   0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

Links

Crossrefs

Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.
Main diagonal gives A092366.
Cf. A107267, A292627, A307819, A307847, A307855, A307883.

Programs

  • Mathematica
    A[n_, k_] := k^n Hypergeometric2F1[(1-n)/2, -n/2, 1, 4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

Formula

A(n,k) is the coefficient of x^n in the expansion of (1 + k*x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = k * (2*n-1) * A(n-1,k) - k * (k-4) * (n-1) * A(n-2,k).

A307844 Constant term in the expansion of (n/x + 1 + n*x)^n.

Original entry on oeis.org

1, 1, 9, 55, 1729, 19251, 1050841, 16977129, 1322929665, 28017221059, 2839212609001, 74390784295653, 9283240524317761, 289865990675075725, 42976734096778661817, 1557837326400792009751, 267561369300137776050177, 11042876765198762014337235
Offset: 0

Views

Author

Seiichi Manyama, May 01 2019

Keywords

Comments

Also coefficient of x^n in the expansion of (1 + x + (n*x)^2)^n.
Also coefficient of x^n in the expansion of 1/sqrt(1 - 2*x + (1-4*n^2)*x^2).

Links

Crossrefs

Main diagonal of A307847.
Cf. A002426, A092366, A186925, A187018.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[(-1)^k * (2*n + 1)^(n-k) * n^k * Binomial[n,k] * Binomial[2*k,k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 02 2019 *)
  • PARI
    {a(n) = polcoef((n/x+1+n*x)^n, 0)}
  • PARI
    {a(n) = polcoef((1+x+(n*x)^2)^n, n)}
  • PARI
    {a(n) = sum(k=0, n, (1-2*n)^(n-k)*n^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n, (1+2*n)^(n-k)*(-n)^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n\2, n^(2*k)*binomial(n, 2*k)*binomial(2*k, k))}

Formula

a(n) = Sum_{k=0..n} (1-2*n)^(n-k) * n^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (1+2*n)^(n-k) * (-n)^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..floor(n/2)} n^(2*k) *binomial(n,2*k) * binomial(2*k,k).
a(n) ~ (exp(1/2) + (-1)^n * exp(-1/2)) * 2^(n - 1/2) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, May 02 2019
Showing 1-4 of 4 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.