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 1-5 of 5 results.

A049209 a(n) = -Product_{k=0..n} (7*k-1); sept-factorial numbers.

Original entry on oeis.org

1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 155008339200, 9610517030400, 663125675097600, 50397551307417600, 4182996758515660800, 376469708266409472000, 36517561701841718784000, 3797826416991538753536000, 421558732286060801642496000
Offset: 0

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Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034833, A045754, A048994, A051188, A084947, A144739, A144827, A147585.
Row sums of triangle A051186 (scaled Stirling1 triangle).
Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), this sequence (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Programs

  • Magma
    [ -&*[ (7*k-1): k in [0..n-1] ]: n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008
  • Mathematica
    CoefficientList[Series[(1-7*x)^(-6/7),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
With[{m=7}, Table[m^n*Pochhammer[(m-1)/m, n], {n, 0, 30}]] (* G. C. Greubel, Feb 16 2022 *)
  • Sage
    m=7; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

Formula

a(n) = 6*A034833(n) = (7*n-1)*(!^7), n >= 1, a(0) := 1.
a(n) = Product_{k=1..n} (7*k - 1). a(0) = 1; a(n) = (7*n - 1)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008
G.f.: 1/(1-6*x/(1-7*x/(1-13*x/(1-14*x/(1-20*x/(1-21*x/(1-27*x/(1-28*x/(1-...(continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 7^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = 7^n * Gamma(n+6/7) / Gamma(6/7). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-7*x)^(-6/7). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (14*k+6)*x - 7*(k+1)*(7*k+6)*x^2/G(k+1); (continued fraction).
which starts as 1/(1-6*x-42*x^2/(1-20*x-182*x^2/(1-34*x-420*x^2/(1-48*x-756*x^2/(1-62*x-1190*x^2/(1-... )))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - (7*k+6)*x/(1 - (7*k+7)*x/Q(k+1) ); (continued fraction). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 19 2022

A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

Original entry on oeis.org

1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200, 9045631231436800, 886471860680806400, 94852489092846284800, 11002888734770169036800, 1375361091846271129600000, 184298386307400331366400000, 26354669241958247385395200000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), this sequence (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).
Cf. A035022, A048994.

Programs

  • Magma
    m:=9; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022
  • Mathematica
    CoefficientList[Series[(1-9*x)^(-8/9),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
  • PARI
    a(n) = prod(k=1, n, 9*k-1); \\ Michel Marcus, Jan 08 2015
  • Sage
    m=9; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

Formula

a(n) = 8*A035022(n) = (9*n-1)(!^9), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 9^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = 9^n * Gamma(n+8/9) / Gamma(8/9). - Vaclav Kotesovec, Jan 28 2015
E.g.f: (1-9*x)^(-8/9). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 09 2020: (Start)
G.f.: 1/(1-8*x-72*x^2/(1-26*x-306*x^2/(1-44*x-702*x^2/(1-62*x-1260*x^2/(1-80*x-1980*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/(1-8*x/(1-9*x/(1-17*x/(1-18*x/(1-26*x/(1-27*x/(1-35*x/(1-36*x/(1-44*x/(1-45*x/(1-...))))))))))) (Stieltjes continued fraction). (End)
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (18*k+8)*x - 9*(k+1)*(9*k+8)*x^2/G(k+1) (continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(9*k+8)/(1 - x*(9*k+9)/Q(k+1) ) (continued fraction). (End)
G.f.: hypergeometric2F0([1, 8/9], [--], 9*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). - Amiram Eldar, Dec 21 2022

Extensions

a(9) (originally given incorrectly as 1011636305971200) corrected by Peter Bala, Feb 20 2015
a(15)-a(16) from Vincenzo Librandi, Feb 20 2015
a(16) corrected and incorrect MAGMA program removed by Georg Fischer, May 10 2021

A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

Original entry on oeis.org

1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 33760034483175, 2667042724170825, 232032717002861775, 22043108115271868625, 2270440135873002468375, 252018855081903273989625, 29990243754746489604765375, 3808760956852804179805202625
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034975, A048994, A051187.
Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), this sequence (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Programs

  • Magma
    m:=8; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..30]]; // G. C. Greubel, Feb 16 2022
  • Mathematica
    FoldList[Times,1,8*Range[20]-1] (* Harvey P. Dale, Aug 03 2014 *)
CoefficientList[Series[(1-8*x)^(-7/8),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
  • PARI
    a(n) = -prod(k=0, n, 8*k-1); \\ Michel Marcus, Jan 08 2015
  • Sage
    m=8; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

Formula

a(n) = 7*A034975(n) = (8*n-1)(!^8), n >= 1, a(0) = 1.
G.f.: 1/(1-7*x/(1-8*x/(1-15*x/(1-16*x/(1-23*x/(1-24*x/(1-31*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-1)^n*Sum_{k=0..n} 8^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: ( 1 - 1/Q(0) )/x where Q(k) = 1 - x*(8*k-1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = 8^n*Gamma(n+7/8)/Gamma(7/8). - R. J. Mathar, Mar 20 2013
E.g.f: (1-8*x)^(-7/8). - Vaclav Kotesovec, Jan 28 2015
G.f.: 1/(1-7*x-56*x^2/(1-23*x-240*x^2/(1-39*x-552*x^2/(1-55*x-992*x^2/(1-71*x-1560*x^2/(1-... )))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Dec 09 2020
G.f.: 1/G(0) where G(k) = 1 - (16*k+7)*x - 8*(k+1)*(8*k+7)*x^2/G(k+1); (continued fraction). - Nikolaos Pantelidis, Dec 19 2020
Sum_{n>=0} 1/a(n) = 1 + (e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). - Amiram Eldar, Dec 20 2022

A254322 Expansion of e.g.f.: (1-11*x)^(-10/11).

Original entry on oeis.org

1, 10, 210, 6720, 288960, 15603840, 1014249600, 77082969600, 6706218355200, 657209398809600, 71635824470246400, 8596298936429568000, 1126115160672273408000, 159908352815462823936000, 24465977980765812062208000, 4012420388845593178202112000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 28 2015

Keywords

Comments

Generally, for k > 1, if e.g.f. = (1-k*x)^(-(k-1)/k) then a(n) ~ n! * k^n / (n^(1/k) * Gamma((k-1)/k)).

Links

Crossrefs

Sequences of the form k^n*Pochhammer((k-1)/k, n): A000007 (k=1), A001147 (k=2), A008544 (k=3), A008545 (k=4), A008546 (k=5), A008543 (k=6), A049209 (k=7), A049210 (k=8), A049211 (k=9), A049212 (k=10), this sequence (k=11), A346896 (k=12).
Cf. A000108, A254282, A254286, A254287.

Programs

  • Magma
    m=11; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022
  • Mathematica
    CoefficientList[Series[(1-11*x)^(-10/11), {x, 0, 20}], x] * Range[0, 20]!
FullSimplify[Table[11^n * Gamma[n+10/11] / Gamma[10/11], {n, 0, 18}]]
  • Sage
    m=11; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

Formula

D-finite with recurrence: a(0) = 1; a(n) = (11*n-1) * a(n-1) for n > 0. [corrected by Georg Fischer, Dec 23 2019]
a(n) = 11^n * Gamma(n+10/11) / Gamma(10/11).
a(n) ~ n! * 11^n / (n^(1/11) * Gamma(10/11)).
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (22*k+10)*x - 11*(k+1)*(11*k+10)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-10*x-110*x^2/(1-32*x-462*x^2/(1-54*x-1056*x^2/(1-76*x-1892*x^2/(1-98*x-2970*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(11*k+10)/(1 - x*(11*k+11)/Q(k+1)) (continued fraction).
G.f.: 1/(1-10*x/(1-11*x/(1-21*x/(1-22*x/(1-32*x/(1-33*x/(1-43*x/(1-44*x/(1-54*x/(1-55*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: hypergeometric2F0([1, 10/11], [--], 11*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/11)^(1/11)*(Gamma(10/11) - Gamma(10/11, 1/11)). - Amiram Eldar, Dec 22 2022

A354394 Expansion of e.g.f. 1/(1 + (exp(x) - 1)^5 / 120).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -15, -140, -1050, -6951, -42273, -232870, -949740, 2401399, 149618469, 2979464124, 47639256210, 683529622229, 9045426379611, 109599657976942, 1148191101672384, 8033814119097459, -50834295574038207, -3977581842278623216, -119536187842156328034
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Cf. A354391, A354392, A354393.
Cf. A346896, A346924, A354398.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^5/120)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i, j)*stirling(j, 5, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/(-120)^k);

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(-120)^k.
Showing 1-5 of 5 results.

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