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.

A091547 Row sums of triangle A091543.

Original entry on oeis.org

1, 3, 11, 93, 1837, 64543, 3754807, 364026729, 57155586089, 13507554452859, 4491954229297987, 2028035990665237525, 1227168186637813749445, 987585965101526606515191, 1041500427833355704666149103
Offset: 1

Views

Author

Wolfdieter Lang, Feb 13 2004

Keywords

Formula

a(n)=sum(A091543(n, m), m=0..n-1), n>=1.

A091548 Alternating row sums of triangle A091543.

Original entry on oeis.org

1, 1, -1, -53, -1083, -24099, -120477, 115958039, 30323462809, 7554183583865, 2000735772375143, 483441809221224835, -15004002104132799283, -243176330287496140346187, -424738050855195910944014901, -660294012298780945217254277745
Offset: 1

Views

Author

Wolfdieter Lang, Feb 13 2004

Keywords

Formula

a(n)=sum(A091543(n, m)*(-1)^m, m=0..n-1), n>=1.

A091544 First column sequence of array A091746 ((6,2)-Stirling2).

Original entry on oeis.org

1, 30, 2700, 491400, 150368400, 69470200800, 45155630520000, 39285398552400000, 44078217175792800000, 61973973349164676800000, 106719182107261573449600000, 220908706962031457040672000000, 541226332056977069749646400000000, 1548989762347068373623487996800000000
Offset: 1

Views

Author

Wolfdieter Lang, Feb 13 2004

Keywords

Comments

Also fifth column (m=4) sequence of triangle A091543.

Links

Crossrefs

Cf. A091535 (third column of A091543, first column of array A091534), A000407, A007696, A091746.

Programs

  • Mathematica
    a[n_] := 2^(4*n-1) * Pochhammer[1/4, n] * Pochhammer[1/2, n]; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

Formula

a(n) = 2^(n-1)*Product_{j=0..n-1}((2*j+1)*(4*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=1.
a(n) = (2^(4*n-1))*risefac(1/4, n)*risefac(1/2, n), n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac4(4*n-3)*fac4(4*n-2)/2, n>=1, with fac4(4*n-3) = A007696(n) and fac4(4*n-2)/2 = A000407(n+1) (quartic- or 4-factorials).
E.g.f.: (hypergeom([1/4, 1/2], [], 16*x)-1)/2.
a(n) = A091746(n, 2), n>=1.
a(n) ~ sqrt(Pi) * 2^(4*n) * n^(2*n-1/4) / (Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025

A091545 First column sequence of the array (7,2)-Stirling2 A091747.

Original entry on oeis.org

1, 42, 5544, 1507968, 696681216, 489070213632, 485157651922944, 646229992361361408, 1112808046846264344576, 2405890997281623512973312, 6380422924790865556405223424, 20366309975932442856045473169408, 77025384328976498881563979526701056, 340606249502734078054275917467072069632
Offset: 1

Views

Author

Wolfdieter Lang, Feb 13 2004

Keywords

Comments

Also sixth column (m=5) sequence of triangle A091543.

Links

Crossrefs

Cf. A008548, A034323, A091543, A091747, A175380, A246745.

Programs

  • Mathematica
    a[n_] := 5^(2*n) * Pochhammer[1/5, n] * Pochhammer[2/5, n] / 2; Array[a, 15] (* Amiram Eldar, Sep 01 2025 *)

Formula

a(n) = Product_{j=0..n-1} ((5*j+2)*(5*j+1))/2, n>=1. From eq.12 of the Blasiak et al. reference with r=7, s=2, k=1.
a(n) = (5^(2*n))*risefac(1/5, n)*risefac(2/5, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac5(5*n-3)*fac5(5*n-4)/2, n>=1, with fac5(5*n-4)/2 = A034323(n) and fac5(5*n-3) = A008548(n) (5-factorials).
E.g.f.: (hypergeom([1/5, 2/5], [], 25*x)-1)/2.
a(n) = A091747(n, 2), n>=1.
D-finite with recurrence a(n) - (5*n-3)*(5*n-4)*a(n-1) = 0. - R. J. Mathar, Jul 27 2022
a(n) ~ Pi * (5/e)^(2*n) * n^(2*n-2/5) / (Gamma(1/5) * Gamma(2/5)). - Amiram Eldar, Sep 01 2025
a(n) ~ sqrt(Pi*(1 + sqrt(5))) * 5^(2*n + 1/4) * n^(2*n - 2/5) / (Gamma(1/10) * 2^(7/10) * exp(2*n)). - Vaclav Kotesovec, Sep 01 2025

A091546 First column of the array A092077 ((8,2)-Stirling2).

Original entry on oeis.org

1, 56, 10192, 3872960, 2517424000, 2497284608000, 3511182158848000, 6643156644540416000, 16275733779124019200000, 50129260039701979136000000, 189588861470152885092352000000, 863766852858016544480755712000000, 4666068539139005373285042356224000000, 29489553167358513959161467691335680000000
Offset: 1

Views

Author

Wolfdieter Lang, Feb 13 2004

Keywords

Comments

Also seventh column (m=6) of triangle A091543.

Links

Crossrefs

Cf. A007559, A008542, A034689, A091543, A092077.
Cf. A073005, A175379.

Programs

  • Mathematica
    a[n_] := 6^(2*n) * Pochhammer[1/6, n] * Pochhammer[1/3, n] / 2; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

Formula

a(n) = (2^(n-1))*Product_{j=0..n-1} ((3*j+1)*(6*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=8, s=2, k=1.
a(n) = (6^(2*n))*risefac(1/6, n)*risefac(1/3, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac6(6*n-5)*fac6(6*n-4)/2, n>=1, with fac6(6*n-5) = A008542(n) and fac6(6*n-4)/2 = A034689(n)= (2^(n-1))*A007559(n), (6-factorials).
a(n) ~ Pi * (6/e)^(2*n) * n^(2*n-1/2) / (Gamma(1/6) * Gamma(1/3)). - Amiram Eldar, Aug 30 2025
Showing 1-5 of 5 results.

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