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 1-10 of 12 results. Next

A365752 Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^4 ).

Original entry on oeis.org

1, 3, 16, 103, 735, 5592, 44452, 364815, 3067558, 26290517, 228819168, 2016953848, 17968790029, 161536295244, 1463535347928, 13349907110367, 122499957767130, 1130001670577730, 10472708110616136, 97468774074103041, 910582642690819351
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2023

Keywords

Links

Crossrefs

Cf. A063020, A365751, A365753.
Cf. A365856, A368079.
Cf. A365754, A370105.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(5*n-k+3, n-k))/(n+1);
  • SageMath
    def A365752(n):
    h = binomial(5*n + 3, n) * hypergeometric([-n, n + 1], [-5 * n - 3], -1) / (n + 1)
    return simplify(h)
print([A365752(n) for n in range(21)])  # Peter Luschny, Sep 20 2023

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(5*n-k+3,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(4*n-2*k+2,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x) * (1-x)^4 )^(n+1). - Seiichi Manyama, Feb 16 2024

A365755 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x)^5 ).

Original entry on oeis.org

1, 6, 52, 530, 5919, 70098, 864784, 10994490, 143042020, 1895316632, 25487708844, 346976558318, 4772478619146, 66222166440780, 925880434336320, 13030945427540170, 184467676431001644, 2624828100099166536, 37521220349342729680, 538573138719587026440
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2023

Keywords

Crossrefs

Cf. A000108, A006318, A107111, A263843, A365754.
Cf. A365753.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+k, k)*binomial(5*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(5*(n+1),n-k).

A365847 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ).

Original entry on oeis.org

1, 8, 96, 1368, 21440, 356968, 6197408, 110947768, 2033381760, 37963483592, 719495148768, 13806129179928, 267693334199616, 5236670783633960, 103227182363423008, 2048451544990578552, 40888361539777714944, 820400146864231266184
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A066357, A365754, A365846, A365848.
Cf. A032349, A365622, A365843.
Cf. A260332.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*n+k+3, k)*binomial(4*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(4*(n+1),n-k).
Conjecture: g.f.: B^4, where B is the g.f. of A260332.

A366016 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 4 * A(x)).

Original entry on oeis.org

0, 1, 8, 102, 1580, 27193, 499828, 9609372, 190869948, 3886281300, 80681111940, 1701418017390, 36345240847188, 784821812522062, 17103169093916120, 375670490644949624, 8308349385885678684, 184856293637482503660, 4134886240989315235840, 92928784113832360511800, 2097399158679611824619120
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

Comments

Reversion of g.f. for heptagonal pyramidal numbers (with signs).

Links

Crossrefs

Cf. A002293, A002413, A064088, A365754, A365817, A366014, A366015, A366017.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 4 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 4^k, {k, 0, n - 1}], {n, 1, 20}]]

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 4^k for n > 0.
a(n) ~ sqrt(163 - 1521/sqrt(89)) * (4933 + 801*sqrt(89))^n / (sqrt(Pi) * n^(3/2) * 2^(9*n + 9/2)). - Vaclav Kotesovec, Sep 27 2023

A366014 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 2 * A(x)).

Original entry on oeis.org

0, 1, 6, 54, 580, 6873, 86688, 1141500, 15512220, 215928900, 3063184410, 44124882750, 643692232404, 9490176205006, 141184118174640, 2116751269990968, 31951313566227228, 485159929343783532, 7405637373574690968, 113572576254948487800, 1749075343256441443320
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

Comments

Reversion of g.f. for pentagonal pyramidal numbers (with signs).

Links

Crossrefs

Cf. A002293, A002411, A064063, A365754, A366015, A366016, A366017.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 2 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 2^k, {k, 0, n - 1}], {n, 1, 20}]]

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 2^k for n > 0.

A366015 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 7, 76, 995, 14433, 223300, 3611016, 60305787, 1032115315, 18007816255, 319110233104, 5727667197044, 103913426353324, 1902498385538520, 35106179258551632, 652236828560562987, 12190651925663309175, 229059610932456616501, 4324334144117016053500, 81983637468108446363755
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

Comments

Reversion of g.f. for hexagonal pyramidal numbers (with signs).

Links

Crossrefs

Cf. A002293, A002412, A064087, A365754, A365816, A366014, A366016, A366017.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 3^k for n > 0.

A366017 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 5 * A(x)).

Original entry on oeis.org

0, 1, 9, 132, 2365, 47169, 1005564, 22431720, 517122117, 12222124035, 294569159313, 7212098118888, 178877944712844, 4484938858752940, 113488477622130600, 2894560146756466320, 74335973069605120725, 1920587845828953301479, 49886703842977713177723, 1301959618949870922531300, 34123873581608909988904245
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

Comments

Reversion of g.f. for octagonal pyramidal numbers (with signs).

Links

Crossrefs

Cf. A002293, A002414, A064089, A365754, A365818, A366014, A366015, A366016.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 5 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 5^k, {k, 0, n - 1}], {n, 1, 20}]]

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 5^k for n > 0.

A365846 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^4 ).

Original entry on oeis.org

1, 7, 73, 903, 12281, 177415, 2672377, 41506823, 659972089, 10689904647, 175765581817, 2925998735367, 49219210772473, 835307328307207, 14284937032826873, 245924997499453447, 4258621314671050745, 74128819286282600455, 1296324135131612708857
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A066357, A365754, A365847, A365848.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+k+2, k)*binomial(4*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(4*(n+1),n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^4 / (1-x)^3 )^(n+1). - Seiichi Manyama, Jul 31 2025

A370100 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 5, 47, 500, 5615, 65005, 767396, 9183144, 110995695, 1351922495, 16566597047, 204010570296, 2522556212228, 31298015910140, 389458822888280, 4858487926378000, 60742838865326319, 760901358321592611, 9547848458062427405, 119990407515367475700
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Crossrefs

Cf. A001448, A352373, A370101, A370102.
Cf. A365754.

Programs

  • Mathematica
    Table[Sum[Binomial[4*n, k]*Binomial[2*n - k - 1, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2024 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n, k)*binomial(2*n-k-1, n-k));

Formula

a(n) = [x^n] ( (1+x)^4/(1-x) )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)/(1+x)^4 ). See A365754.
From Peter Bala, Jun 08 2024: (Start)
2*n*(n - 1)*(2*n - 1)*(51*n^2 - 144*n + 100)*a(n) = -(n - 1)*(5457*n^4 - 20865*n^3 + 26366*n^2 - 12172*n + 1560)*a(n-1) + 64*(2*n - 3)*(4*n - 5)*(4*n - 7)*(51*n^2 - 42*n + 7)*a(n-2) with a(0) = 1 and a(1) = 5.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. See A352373 for a more general conjecture. (End)
a(n) ~ sqrt(3 + 5/sqrt(17)) * (51*sqrt(17) - 107)^n / (sqrt(Pi*n) * 2^(3*n + 3/2)). - Vaclav Kotesovec, Jun 12 2024
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(3*n-k,n-k). (End)

A365848 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^4 ).

Original entry on oeis.org

1, 9, 122, 1965, 34814, 655290, 12861708, 260312853, 5393696150, 113847928558, 2439377254412, 52919446267698, 1160040801590332, 25655668799151700, 571760925292574640, 12827392114274902629, 289470689505615716070, 6566330844138035042982
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A066357, A365754, A365846, A365847.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(4*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(4*(n+1),n-k).
Showing 1-10 of 12 results. Next

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