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.

A383965 Self-convolution square-root of A004381, where A004381(n) = binomial(8*n,n).

Original entry on oeis.org

1, 4, 52, 804, 13412, 233548, 4180932, 76307228, 1412731844, 26443784224, 499310856828, 9494966722696, 181620437132820, 3491268491768400, 67396227598309788, 1305787014634864584, 25380012805871145604, 494684878753394992992, 9665968233663380580256, 189289570996914582016788
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 06 2025

Keywords

Comments

In general, for m > 1, if Sum_{k=0..n} a(k)*a(n-k) = binomial(m*n,n), then a(n) ~ m^(m*n + 1/4) / (2^(1/4) * Gamma(1/4) * (m-1)^((m-1)*n + 1/4) * n^(3/4)).

Crossrefs

Cf. A004381, A007556, A208977, A384695.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      (binomial(8*n, n)-add(a(j)*a(n-j), j=1..n-1))/2)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 06 2025
  • Mathematica
    nmax = 20; self = ConstantArray[0, nmax + 1]; self[[1]] = 1; self[[2]] = 4; Do[self[[k+1]] = (Binomial[8*k, k] - Sum[self[[j+1]]*self[[k - j + 1]], {j, 1, k-1}]) / (2*self[[1]]);, {k, 2, nmax}]; self

Formula

a(n) ~ 2^(24*n + 1/2) / (Gamma(1/4) * 7^(7*n + 1/4) * n^(3/4)).
From Seiichi Manyama, Aug 16 2025: (Start)
Sum_{k=0..n} a(k) * a(n-k) = A004381(n).
G.f.: 1/sqrt(1 - 8*x*g^7) where g = 1+x*g^8 is the g.f. of A007556.
G.f.: sqrt( g/(8-7*g) ) where g = 1+x*g^8 is the g.f. of A007556. (End)

A385719 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)/3), where B(x) is the g.f. of A004355.

Original entry on oeis.org

1, 4, 38, 428, 5204, 66104, 863840, 11515308, 155779966, 2131436392, 29426804398, 409254436452, 5726378247412, 80535621269208, 1137609359823936, 16130112288879248, 229462608491483364, 3273749607191060480, 46826932120849617128, 671341041479214814160, 9644654058165119642624
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2025

Keywords

Links

Crossrefs

Cf. A183161, A208977, A387084, A387086.
Cf. A002295, A004355, A385497.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[6*n, n]*x^n, {n, 0, nmax}] - 1)/3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)

Formula

Sum_{k=0..n} a(k) * a(n-k) = A385497(n).
G.f.: 1/sqrt(1 - 4*x*g^4*(3-g)) where g = 1+x*g^6 is the g.f. of A002295.
G.f.: g/sqrt((2-g) * (6-5*g)) where g = 1+x*g^6 is the g.f. of A002295.
a(n) ~ 2^(6*n - 1/2) * 3^(6*n + 3/4) / (Gamma(1/4) * n^(3/4) * 5^(5*n + 1/4)) * (1 + 7*Gamma(1/4)^2/(48*Pi*sqrt(30*n))). - Vaclav Kotesovec, Aug 20 2025

A387084 Expansion of B(x)/sqrt(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449.

Original entry on oeis.org

1, 3, 23, 211, 2095, 21752, 232439, 2534182, 28041295, 313833025, 3544160216, 40318629754, 461455158383, 5308453068900, 61333295856750, 711305543582150, 8276351877367663, 96576953297406377, 1129842469637643485, 13248082583624602575, 155660344852055352760
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Crossrefs

Cf. A183161, A208977, A387086.
Cf. A001449, A002294, A079589.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Sum[Binomial[5*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 4*(Sum[Binomial[5*n, n]*x^n, {n, 0, nmax}] - 1)/5], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)

Formula

Sum_{k=0..n} a(k) * a(n-k) = A079589(n).
G.f.: 1/sqrt(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g/sqrt(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.
Conjecture D-finite with recurrence 3902464*n*(8*n-5) *(8*n-3)*(8*n-1) *(8*n+1)*a(n) +80*(-12565760000*n^5 +68448000000*n^4 -163457516000*n^3 +200475354000*n^2 -122843089511*n +29804717943)*a(n-1) +125000*(134055000*n^5 -1109795000*n^4 +3726971625*n^3 -6307124125*n^2 +5325821766*n -1769460798)*a(n-2) +48828125*(-1556875*n^5 +15845625*n^4 -60659875*n^3 +103818375*n^2 -67764178*n +1391424)*a(n-3) -152587890625 *(5*n-16)*(n-3) *(5*n-19)*(5*n-18) *(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 19 2025
a(n) ~ 5^(5*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(8*n + 7/4)). - Vaclav Kotesovec, Aug 20 2025

A387086 Expansion of B(x)/sqrt(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.

Original entry on oeis.org

1, 0, 2, 4, 16, 52, 188, 672, 2458, 9052, 33648, 125864, 473500, 1789632, 6791528, 25863568, 98796096, 378411332, 1452886052, 5590262688, 21551271916, 83228809640, 321933018272, 1247062996304, 4837152438556, 18785529571200, 73037938668632, 284268423472432
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Crossrefs

Cf. A183161, A208977, A387084.
Cf. A000108, A000984, A387085.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] / Sqrt[1 + 2*(Sum[Binomial[2*n, n]*x^n, {n, 0, nmax}] - 1)], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(4*x-1+2*sqrt(1-4*x)))

Formula

Sum_{k=0..n} a(k) * a(n-k) = A387085(n).
G.f.: 1/sqrt( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/sqrt(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g/sqrt((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
a(n) ~ 2^(2*n - 1/2) / (Gamma(1/4) * n^(3/4)) * (1 - Gamma(1/4)^2/(16*Pi*sqrt(2*n))). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*(n-1)*a(n) -2*(n-1)*(10*n-17)*a(n-1) +4*(4*n^2-24*n+29)*a(n-2) +32*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

A384695 Self-convolution square-root of A169961, where A169961(n) = binomial(12*n,n).

Original entry on oeis.org

1, 6, 120, 2850, 72990, 1950816, 53594508, 1500996420, 42639593040, 1224606404670, 35477155257720, 1035058071490152, 30375294227227530, 895810786837337880, 26530164526824124560, 788575111385154710700, 23513904388397505712014, 703104985574123730695460, 21076207836773295148694400
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 07 2025

Keywords

Crossrefs

Cf. A169961, A208977, A383965.

Programs

  • Mathematica
    nmax = 20; self = ConstantArray[0, nmax + 1]; self[[1]] = 1; self[[2]] = 6; Do[self[[k+1]] = (Binomial[12*k, k] - Sum[self[[j+1]]*self[[k - j + 1]], {j, 1, k-1}]) / (2*self[[1]]);, {k, 2, nmax}]; self

Formula

a(n) ~ 12^(12*n + 1/4) / (2^(1/4) * Gamma(1/4) * 11^(11*n + 1/4) * n^(3/4)).
From Seiichi Manyama, Aug 16 2025: (Start)
Sum_{k=0..n} a(k) * a(n-k) = A169961(n).
G.f.: 1/sqrt(1 - 12*x*g^11) where g = 1+x*g^12.
G.f.: sqrt( g/(12-11*g) ) where g = 1+x*g^12. (End)
Showing 1-5 of 5 results.

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