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.

A355372 Expansion of the e.g.f. log((1 - x) / (1 - 2*x)) / (1 - x)^3.

Original entry on oeis.org

0, 1, 9, 77, 714, 7374, 85272, 1102968, 15908400, 254866320, 4516084800, 88102382400, 1883199024000, 43885950595200, 1109416142822400, 30273281955302400, 887493144729139200, 27827941161784780800, 929449073791558656000, 32943696020637889536000, 1234946945823695419392000
Offset: 0

Views

Author

Mélika Tebni, Jun 30 2022

Keywords

Comments

Conjecture: For p prime, a(p) == -1 (mod p).

Crossrefs

Cf. A000292, A062139, A355171.

Programs

  • Maple
    A355372 := n -> A000292(n)*n!*hypergeom([1 - n, 1, 1], [2, 4], -1):
seq(simplify(A355372(n)), n = 0..20);
  • Mathematica
    CoefficientList[Series[Log[(1 - x)/(1 - 2*x)]/ (1 - x)^3,{x,0,20}],x]Table[n!,{n,0,20}] (* Stefano Spezia, Jun 30 2022 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A062139(n, k + 1).
a(0) = 0, a(n) = n!*Sum_{k=1..n} (n-k+2)*(n-k+1)*(2^k-1)/(2*k).
a(n) = A000292(n)*n!*hypergeom([1 - n, 1, 1], [2, 4], -1). - Peter Luschny, Jun 30 2022

A355257 Array read by ascending antidiagonals. A(n, k) = k! * [x^k] log((1 - x) / (1 - 2*x)) / (1 - x)^n, for 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 14, 0, 1, 7, 29, 90, 0, 1, 9, 50, 206, 744, 0, 1, 11, 77, 406, 1774, 7560, 0, 1, 13, 110, 714, 3804, 18204, 91440, 0, 1, 15, 149, 1154, 7374, 41028, 218868, 1285200, 0, 1, 17, 194, 1750, 13144, 85272, 506064, 3036144, 20603520
Offset: 0

Views

Author

Peter Luschny and Mélika Tebni, Jul 01 2022

Keywords

Comments

Conjecture: For p prime, A(n, p) == -1 (mod p) for n >= 0.
Conjecture: Let n >= 0, k >= 1 and k != 4. Then k divides A(n, k) if and only if k is not prime.
From Mélika Tebni, Jul 04 2022: (Start)
Conjecture: The polynomials of A355259 generate the k+1 column of this array.
Conjecture: For p prime and n even, (A(n, p) / (p - 1)) == 1 (mod p). (End)

Examples

			Table A(n, k) begins:
  [0] 0, 1,  3,  14,   90,   744,   7560,    91440,   1285200, ... A029767
  [1] 0, 1,  5,  29,  206,  1774,  18204,   218868,   3036144, ... A103213
  [2] 0, 1,  7,  50,  406,  3804,  41028,   506064,   7084656, ... A355171
  [3] 0, 1,  9,  77,  714,  7374,  85272,  1102968,  15908400, ... A355372
  [4] 0, 1, 11, 110, 1154, 13144, 164136,  2251920,  33923760, ... A355407
  [5] 0, 1, 13, 149, 1750, 21894, 295500,  4320420,  68487120, ... A355414
  [6] 0, 1, 15, 194, 2526, 34524, 502644,  7838928, 131198544, ...
  [7] 0, 1, 17, 245, 3506, 52054, 814968, 13543704, 239548176, ...

Crossrefs

Cf. A029767, A103213, A355171, A355259, A355372, A355407, A355414.

Programs

  • Maple
    egf := n -> log((1 - x)/(1 - 2*x))/(1 - x)^n:
ser := n -> series(egf(n), x, 22):
row := n -> seq(k!*coeff(ser(n), x, k), k = 0..8):
seq(print(row(n)), n = 0..8);
# Alternative:
A := (n, k) -> add(k!*binomial(k + n - 1, k - j - 1)/(j + 1), j = 0..k-1):
seq(print(seq(A(n, k), k = 0..8)), n = 0..7);
  • Mathematica
    A[0, 0] = 0; A[n_, k_] := k! * Binomial[n+k-1, k - 1] * HypergeometricPFQ[{1 - k, 1, 1}, {2, n + 1}, -1];
Table[A[n, k], {n, 0, 8}, {k, 0, 8}] // TableForm

Formula

A(n, k) = k!*Sum_{j=0..k-1} binomial(k + n - 1, k - j - 1) / (j + 1).
A(n, k) = k!*Sum_{j=1..k} binomial(n + k - j - 1, n - 1)*(2^j - 1) / j.
A(n, k) = k!*binomial(n + k - 1, k - 1)*hypergeom([1, 1, 1 - k], [2, n + 1], -1) except for A(0, 0) = 0.

A355407 Expansion of the e.g.f. log((1 - x) / (1 - 2*x)) / (1 - x)^4.

Original entry on oeis.org

0, 1, 11, 110, 1154, 13144, 164136, 2251920, 33923760, 560180160, 10117886400, 199399132800, 4275988617600, 99473802624000, 2502049379558400, 67804022648678400, 1972357507107993600, 61358018782620672000, 2033893411878730752000, 71587670846333773824000, 2666700362750370895872000
Offset: 0

Views

Author

Mélika Tebni, Jul 01 2022

Keywords

Comments

Conjecture: For p prime, a(p) == -1 (mod p).

Crossrefs

Cf. A000292, A000332, A062137, A355171, A355372.

Programs

  • Maple
    egf := log((1 - x)/(1 - 2*x))/(1 - x)^4: ser := series(egf, x, 22):
seq(n!*coeff(ser, x, n), n = 0..20); # Peter Luschny, Jul 01 2022
  • Mathematica
    With[{nn=20},CoefficientList[Series[Log[((1-x)/(1-2x))]/(1-x)^4,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Mar 09 2023 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A062137(n, k+1).
a(0) = 0, a(n) = n!*Sum_{k=1..n} A000292(n-k+1)*(2^k-1)/k.
a(n) = A000332(n+3)*n!*hypergeom([1 - n, 1, 1], [2, 5], -1). - Peter Luschny, Jul 01 2022

A355414 Expansion of the e.g.f. log((1 - x) / (1 - 2*x)) / (1 - x)^5.

Original entry on oeis.org

0, 1, 13, 149, 1750, 21894, 295500, 4320420, 68487120, 1176564240, 21883528800, 440117949600, 9557404012800, 223720054790400, 5634130146624000, 152315974848038400, 4409413104676608000, 136318041562123008000, 4487618159996944896000, 156852415886275726848000, 5803748680475885432832000
Offset: 0

Views

Author

Mélika Tebni, Jul 01 2022

Keywords

Comments

Conjecture: For p prime, a(p) == -1 (mod p).

Crossrefs

Cf. A000332, A062140, A355171, A355372, A355407.

Programs

  • Maple
    A355414 := proc(n)
    n!*binomial(n+4,5)*hypergeom([1-n,1,1],[2,6],-1) ;
    simplify(%) ;
end proc:
seq(A355414(n),n=0..40) ; # R. J. Mathar, Jul 27 2022
  • Mathematica
    With[{nn=20},CoefficientList[Series[Log[((1-x)/(1-2x))]/(1-x)^5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 02 2025 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A062140(n, k+1).
a(0) = 0, a(n) = n!*Sum_{k=1..n} A000332(n-k+4)*(2^k-1)/k.
a(n) = binomial(n+4, 5)*n!*hypergeom([1 - n, 1, 1], [2, 6], -1). - Peter Luschny, Jul 01 2022
D-finite with recurrence a(n) +(-4*n-5)*a(n-1) +(n+3)*(5*n-3)*a(n-2) -2*(n-2)*(n+3)*(n+2)*a(n-3)=0. - R. J. Mathar, Jul 27 2022
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.