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.

A216703 a(n) = Product_{k=1..n} (49 - 7/k).

Original entry on oeis.org

1, 42, 1911, 89180, 4213755, 200574738, 9594158301, 460519598448, 22162505675310, 1068725273676060, 51619430718553698, 2496503376570051576, 120872371815599997138, 5857661095679076784380, 284096563140435224042430, 13788153197749122873525936
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A004994, A049382, A216702, A220607.
Cf. A034835, A386271, A386272, A386273, A386274.

Programs

  • Maple
    seq(product(49-7/k, k=1.. n), n=0..20);
seq((7^n/n!)*product(7*k+6, k=0.. n-1), n=0..20);
  • Mathematica
    Table[49^n * Pochhammer[6/7, n] / n!, {n, 0, 15}] (* Amiram Eldar, Aug 17 2025 *)

Formula

From Seiichi Manyama, Jul 17 2025: (Start)
G.f.: 1/(1 - 49*x)^(6/7).
a(n) = (-49)^n * binomial(-6/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+6). (End)
From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 49^n * Gamma(n+6/7) / (Gamma(6/7) * Gamma(n+1)).
a(n) ~ c * 49^n / n^(1/7), where c = 1/Gamma(6/7) = 1/A220607 = 0.904349... . (End)

A386271 Expansion of 1/(1 - 49*x)^(2/7).

Original entry on oeis.org

1, 14, 441, 16464, 662676, 27832392, 1201431588, 52862989872, 2359010923038, 106417603861492, 4842000975697886, 221851681068339504, 10223664969232645476, 473434331652157890504, 22014696421825341908436, 1027352499685182622393680, 48092938891512611510804145
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A020918 (k=2, m=7), A020920 (k=2, m=9), A034835 (k=7, m=1), A034977 (k=8, m=1), A035024 (k=9, m=1), A216702 (k=4, m=3), A216703 (k=7, m=6), A354019 (k=6, m=1), this sequence (k=7, m=2), A386272 (k=7, m=3), A386273 (k=7, m=4), A386274 (k=7, m=5).

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(2/7))

Formula

a(n) = (-49)^n * binomial(-2/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+2).
a(n) = 7^n * Product_{k=1..n} (7 - 5/k).
In general, 1/(1 - k^2*x)^(m/k) leads to the D-finite recurrence k*(k*n-k+m)*a(n-1) - n*a(n) = 0. This sequence is case k=7, m=2: (49*n-35)*a(n-1) - n*a(n) = 0. - Georg Fischer, Jul 19 2025

A386272 Expansion of 1/(1 - 49*x)^(3/7).

Original entry on oeis.org

1, 21, 735, 29155, 1224510, 53143734, 2356038874, 106021749330, 4823989594515, 221367522503855, 10227179539678101, 475098976797773601, 22171285583896101380, 1038639455430209672340, 48816054405219854599980, 2300863364299362480145724, 108715793963144877186885459
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A034835, A216703, A386271, A386273, A386274.
Cf. A049381, A216702.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(3/7))

Formula

a(n) = (-49)^n * binomial(-3/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+3).
a(n) = 7^n * Product_{k=1..n} (7 - 4/k).
D-finite with recurrence n*a(n) +7*(-7*n+4)*a(n-1)=0. - R. J. Mathar, Jul 20 2025

A386273 Expansion of 1/(1 - 49*x)^(4/7).

Original entry on oeis.org

1, 28, 1078, 45276, 1980825, 88740960, 4037713680, 185734829280, 8613452707860, 401961126366800, 18851976826602920, 887756726925482960, 41946505347229069860, 1987619022607162079520, 94411903573840198777200, 4494006610114793461794720, 214307940219849213209335710
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A034835, A216703, A386271, A386272, A386274.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(4/7))

Formula

a(n) = (-49)^n * binomial(-4/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+4).
a(n) = 7^n * Product_{k=1..n} (7 - 3/k).
D-finite with recurrence n*a(n) +7*(-7*n+3)*a(n-1)=0. - R. J. Mathar, Jul 20 2025
Showing 1-4 of 4 results.

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