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 1-7 of 7 results.

A365817 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 4 * A(x)).

Original entry on oeis.org

0, 1, 7, 80, 1119, 17437, 290532, 5066364, 91311055, 1687341227, 31797227631, 608727899936, 11805599569092, 231454163924700, 4579765707561240, 91340133073920420, 1834295500622405295, 37059418988408887015, 752741444501505866325, 15362331852042084534240, 314860558967057266779495
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2023

Keywords

Comments

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

Links

Crossrefs

Cf. A000566, A064088, A113207, A179848, A263843, A365816, A365818.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^3/(1 - 4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 4 x)/(1 + x)^3, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 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(3*n,n-k-1) * 4^k for n > 0.
a(n) ~ 5 * (81 + 21*sqrt(21))^n / (sqrt((427 + 93*sqrt(21))*Pi) * n^(3/2) * 2^(3*n + 3/2)). - Vaclav Kotesovec, Sep 26 2023

A365818 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 5 * A(x)).

Original entry on oeis.org

0, 1, 8, 107, 1760, 32298, 634128, 13034247, 276943568, 6033834950, 134069957840, 3026476515790, 69213144181888, 1600157697995092, 37337615574348960, 878166685063548639, 20797051344280763184, 495509950454603339310, 11869278747340342255440, 285669061791469915886250, 6904850429493240677872320
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2023

Keywords

Comments

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

Links

Crossrefs

Cf. A000567, A064089, A113207, A179848, A263843, A365816, A365817.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^3/(1 - 5 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^3, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 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(3*n,n-k-1) * 5^k for n > 0.
a(n) ~ 3^(3/2) * 2^(n - 1/2) * (154 + 31*sqrt(31))^n / (sqrt((2821 + 506*sqrt(31))*Pi) * n^(3/2) * 5^(2*n)). - Vaclav Kotesovec, Sep 26 2023

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.

A366035 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 8, 98, 1440, 23389, 404712, 7314724, 136476912, 2608808180, 50828498336, 1005682252458, 20152470321984, 408149824237302, 8341496306085040, 171812412714350280, 3562961488550366480, 74328284438252301996, 1558783863783469298016, 32844108784368485209320, 694957689921176181019520
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

Comments

Reversion of g.f. for 4-dimensional figurate numbers A002417 (with signs).

Links

Crossrefs

Cf. A002294, A002417, A064087, A365668, A365755, A365816, A366015, A366036, A366037.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 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(5*n,n-k-1) * 3^k for n > 0.
a(n) ~ 2^(4*n - 1) * 5^(5*n + 1/2) / (sqrt(Pi) * n^(3/2) * 3^(7*n + 5/2)). - Vaclav Kotesovec, Sep 27 2023

A366038 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k) * n^k.

Original entry on oeis.org

1, 2, 25, 658, 27193, 1548526, 112916830, 10062563610, 1061196371665, 129369938790070, 17909387604206371, 2776290021986848588, 476539253976442601735, 89736215305419802692184, 18395742890606906720656524, 4078527943680251523126851306, 972490249766494185823234587681
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

Crossrefs

Cf. A006318, A064063, A135861, A291699, A292798, A365816, A366012, A366016, A366037.

Programs

  • Maple
    A366038 := proc(n)
    add(binomial(n+k,k)*binomial(n*(n+1),n-k)*n^k,k=0..n) ;
    %/(n+1) ;
end proc:
seq(A366038(n),n=0..80) ; # R. J. Mathar, Oct 24 2024
  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[1/(n + 1) Sum[Binomial[n + k, k] Binomial[n (n + 1) , n - k] n^k, {k, 0, n}], {n, 0, 16}]
Table[Binomial[n (n + 1), n] Hypergeometric2F1[-n, n + 1, n^2 + 1, -n]/(n + 1), {n, 0, 16}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - n x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 16}]

Formula

a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - n * x) / (1 + x)^n ).
a(n) ~ phi^(3*n + 3/2) * exp(n/phi^2 + 1/(2*phi)) * n^(n - 3/2) / (5^(1/4) * sqrt(2*Pi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Sep 27 2023

A366203 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * (n-3)^k.

Original entry on oeis.org

1, 2, 12, 156, 3507, 115692, 5066364, 276943568, 18152243967, 1387267590540, 121106707350928, 11889022355301672, 1296359140925188212, 155440199716271334648, 20327081449263918542412, 2879054747404226046119448, 439060192463001381367975215, 71727764882350305085962745740
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 04 2023

Keywords

Comments

a(n) is the coefficient of x^n in expansion of series reversion of g.f. for n-gonal numbers (with signs).

Crossrefs

Cf. A001764, A060354, A113207, A179848, A263843, A323208, A365816, A365817, A365818, A366204.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] (n - 3)^k, {k, 0, n - 1}], {n, 1, 18}]
Table[Binomial[3 n, n - 1] Hypergeometric2F1[1 - n, n, 2 (n + 1), 3 - n]/n, {n, 1, 18}]
Table[SeriesCoefficient[InverseSeries[Series[x (1 - (n - 3) x)/(1 + x)^3, {x, 0, n}], x], {x, 0, n}], {n, 1, 18}]

Formula

a(n) = [x^n] Series_Reversion( x * (1 - (n - 3) * x) / (1 + x)^3 ).

A371394 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+3*x)^3 ).

Original entry on oeis.org

1, 10, 137, 2174, 37562, 686004, 13027065, 254641398, 5089756958, 103552330700, 2137385941418, 44647634773420, 942085264713556, 20050276273007080, 429913404536172633, 9278142975370425510, 201383222768034837750, 4393265621094818733660
Offset: 0

Views

Author

Seiichi Manyama, Mar 21 2024

Keywords

Links

Crossrefs

Cf. A082298, A371393.
Cf. A365816.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+3*x)^3)/x)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(3*(n+1), k)*binomial(2*n-k, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^k * binomial(3*(n+1),k) * binomial(2*n-k,n-k).
Showing 1-7 of 7 results.

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

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