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 1-8 of 8 results.

A016223 Expansion of 1/((1-x) * (1-4*x) * (1-7*x)).

Original entry on oeis.org

1, 12, 105, 820, 6081, 43932, 312985, 2212740, 15576561, 109385452, 767096265, 5375266260, 37649233441, 263634112572, 1845796701945, 12922008569380, 90459786608721, 633241412753292, 4432781515242025, 31029837110570100, 217210325789494401, 1520478144588475612
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A002450, A021874, A111577.
Cf. A282629, A383627.

Programs

  • Maple
    a:=n->sum((7^(n+1-j)-4^(n+1-j))/3, j=0..n+1): seq(a(n), n=0..20); # Zerinvary Lajos, Jan 15 2007
  • PARI
    a(n) = (1-2*4^(n+2)+7^(n+2))/18; \\ Seiichi Manyama, May 03 2025

Formula

a(n) = (1/18) - (16/9)*4^n + (49/18)*7^n. - Antonio Alberto Olivares, Feb 07 2010 [corrected by Seiichi Manyama, May 03 2025]
a(0)=1, a(1)=12, a(n) = 11*a(n-1) - 28*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011
E.g.f.: exp(x)*(1 - 32*exp(3*x) + 49*exp(6*x))/(2!*3^2). - This is (d^2/dx^2) (exp(x)*(exp(x) - 1)^2 / (2*3^2)). See also the second column of the Sheffer triangle A282629 divided by 3^2. - Wolfdieter Lang, Apr 08 2017
From Seiichi Manyama, May 03 2025: (Start)
a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k+2) * Stirling2(k+2,2).
G.f.: B(x)^3, where B(x) is the g.f. of A383627. (End)

A383628 Expansion of 1/( Product_{k=0..3} (1 - (4*k+1) * x) )^(1/4).

Original entry on oeis.org

1, 7, 59, 553, 5555, 58597, 640789, 7201383, 82659891, 964698805, 11408855809, 136374495803, 1644405320701, 19971195162107, 244004256374395, 2996243293813273, 36950056359522771, 457349452121086917, 5678884294812093329, 70710759962448700955, 882616583068179751945
Offset: 0

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Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383629, A383630, A383631, A383632, A383633.
Cf. A383634.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/prod(k=0, 3, 1-(4*k+1)*x)^(1/4))

Formula

a(n) ~ 13^(n + 3/4) / (Gamma(1/4) * 2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 18 2025

A383629 Expansion of 1/( Product_{k=0..4} (1 - (5*k+1) * x) )^(1/5).

Original entry on oeis.org

1, 11, 146, 2156, 34166, 569426, 9854436, 175552696, 3199485331, 59384374841, 1118636310726, 21329345218236, 410804181673996, 7978922735099756, 156074211110053016, 3071360731347145776, 60752572593061028911, 1207041376109801598421, 24073933939936470329806
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383628, A383630, A383631, A383632, A383633.
Cf. A383635.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 4, 1-(5*k+1)*x)^(1/5))

Formula

a(n) ~ 3^(n + 3/5) * 7^(n + 4/5) / (Gamma(1/5) * 2^(3/5) * 5^(4/5) * n^(4/5)). - Vaclav Kotesovec, Aug 18 2025

A383630 Expansion of 1/( Product_{k=0..6} (1 - (7*k+1) * x) )^(1/7).

Original entry on oeis.org

1, 22, 582, 17116, 540457, 17965662, 620869768, 22116614080, 807128297844, 30040462521784, 1136357972482216, 43571763517455888, 1689879290748884068, 66179996449115623096, 2613460738278752421648, 103950807765143954047840, 4160551692685459730727454
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383628, A383629, A383631, A383632, A383633.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 6, 1-(7*k+1)*x)^(1/7))

Formula

a(n) ~ 43^(n + 6/7) / (Gamma(1/7) * 84707280^(1/7) * n^(6/7)). - Vaclav Kotesovec, May 05 2025

A383631 Expansion of 1/( Product_{k=0..7} (1 - (8*k+1) * x) )^(1/8).

Original entry on oeis.org

1, 29, 1009, 39005, 1618849, 70741469, 3214527633, 150606953757, 7231305564225, 354221417305757, 17641204276036657, 890872808134921949, 45521466404971069921, 2349568589682742349405, 122328082368695017498321, 6416984703345086646305181, 338833672698752842286404737
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383628, A383629, A383630, A383632, A383633.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 7, 1-(8*k+1)*x)^(1/8))

Formula

a(n) ~ 3^(n + 5/8) * 19^(n + 7/8) / (Gamma(1/8) * 2^(25/8) * 5^(1/8) * 7^(1/8) * n^(7/8)). - Vaclav Kotesovec, Aug 18 2025

A383632 Expansion of 1/( Product_{k=0..8} (1 - (9*k+1) * x) )^(1/9).

Original entry on oeis.org

1, 37, 1639, 80623, 4257424, 236721412, 13688641144, 816291120808, 49895692924132, 3112177949225236, 197407027057353724, 12699858803178669148, 826900665838817386456, 54398158759680212197576, 3610650035912536155468808, 241521616482786052388206408, 16265890564063100473094045146
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Crossrefs

Cf. A383627, A383628, A383629, A383630, A383631, A383633.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 8, 1-(9*k+1)*x)^(1/9))

Formula

a(n) ~ 4480^(8/9) * 73^(n + 8/9) / (40320 * Gamma(1/9) * n^(8/9)). - Vaclav Kotesovec, Aug 18 2025

A383633 Expansion of 1/( Product_{k=0..10} (1 - (11*k+1) * x) )^(1/11).

Original entry on oeis.org

1, 56, 3741, 277256, 22052713, 1846878936, 160878051401, 14454374710216, 1331486959280259, 125190717874655720, 11973642784650273211, 1161838196321182959096, 114133506709827074843495, 11331528323810252967417064, 1135444330405820622163425351, 114694796036872449398436891896
Offset: 0

Views

Author

Seiichi Manyama, May 03 2025

Keywords

Comments

In general, if m > 0 and g.f. = 1/(Product_{k=0..m-1} (1 - (m*k+1)*x))^(1/m), then a(n) ~ (m*(m-1) + 1)^(n + 1 - 1/m) / (Gamma(1/m) * Gamma(m+1)^(1/m) * m^(1 - 2/m) * n^(1 - 1/m)). - Vaclav Kotesovec, Aug 18 2025

Crossrefs

Cf. A383627, A383628, A383629, A383630, A383631, A383632.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/prod(k=0, 10, 1-(11*k+1)*x)^(1/11))

Formula

a(n) ~ 3^(n + 6/11) * 37^(n + 10/11) / (Gamma(1/11) * 2^(8/11) * 5^(2/11) * 7^(1/11) * 11^(10/11) * n^(10/11)). - Vaclav Kotesovec, May 12 2025

A383935 Expansion of 1 / ( (1-3*x) * (1-6*x) )^(1/3).

Original entry on oeis.org

1, 3, 12, 54, 261, 1323, 6930, 37152, 202554, 1118286, 6233760, 35014356, 197881866, 1123990182, 6411554028, 36705925656, 210797967321, 1213895891835, 7007131607220, 40534622188830, 234931402041525, 1363961443750155, 7931187074571930, 46183636475060760
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2025

Keywords

Links

Crossrefs

Cf. A016137, A383937.
Cf. A383627.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-3*x) * (1-6*x) )^(1/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/((1-3*x)*(1-6*x))^(1/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-1/3, k)*binomial(-1/3, n-k));

Formula

G.f.: B(x)^(1/3), where B(x) is the g.f. of A016137.
a(n) = (-3)^n * Sum_{k=0..n} 2^k * binomial(-1/3,k) * binomial(-1/3,n-k).
a(n) ~ 2^(n + 1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence: (-9*n-3)*a(n)+(18*n-6)*a(n-1)+(n+1)*a(n+1) = 0. - Georg Fischer, Aug 29 2025
Showing 1-8 of 8 results.

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

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