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 10 results.

A307678 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^3/(1 - x).

Original entry on oeis.org

1, 1, 4, 19, 101, 578, 3479, 21714, 139269, 912354, 6078832, 41066002, 280636657, 1936569717, 13475408847, 94446518559, 666149216744, 4724705621702, 33676421377532, 241100485812034, 1732999323835918, 12501487280292424, 90478497094713958, 656788523782034248, 4780725762185300389
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 21 2019

Keywords

Comments

Convolution square root of A270386.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 101*x^4 + 578*x^5 + 3479*x^6 + 21714*x^7 + 139269*x^8 + 912354*x^9 + 6078832*x^10 + ...

Links

Crossrefs

Cf. A001764, A002212, A006013, A127897, A188687 (partial sums), A270386.

Programs

  • Mathematica
    terms = 24; A[] = 1; Do[A[x] = 1 + x A[x]^3/(1 - x) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[Sum[a[k] a[i - k] a[j - i], {k, 0, i}], {i, 0, j}], {j, 0, n - 1}]; Table[a[n], {n, 0, 24}]
terms = 24; CoefficientList[Series[2 Sqrt[(1 - x) Sin[1/3 ArcSin[3/2 Sqrt[3] Sqrt[x/(1 - x)]]]^2/x]/Sqrt[3], {x, 0, terms}], x]
  • Maxima
    a(n):=sum(binomial(n-1,n-k)*(binomial(3*k,k))/(2*k+1),k,0,n); /* Vladimir Kruchinin, Feb 05 2022*/
  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^3, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

a(0) = 1; a(n) = Sum_{j=0..n-1} Sum_{i=0..j} Sum_{k=0..i} a(k)*a(i-k)*a(j-i).
a(n) ~ 31^(n + 1/2) / (3*sqrt(Pi) * n^(3/2) * 2^(2*n+2)). - Vaclav Kotesovec, May 06 2019
G.f.: (2/sqrt(3*x/(1-x)))*sin((1/3)*asin(sqrt((27*x/(1-x))/4))). - Vladimir Kruchinin, Feb 05 2022
a(n) = Sum_{k=0..n} C(n-1,n-k)*C(3*k,k)/(2*k+1). - Vladimir Kruchinin, Feb 05 2022

A371486 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 30, 260, 2465, 24796, 260008, 2811216, 31117240, 350890260, 4016744586, 46556054072, 545273713228, 6443442857024, 76727957438650, 919796418086076, 11091249210406816, 134439965189940176, 1637160457090585016, 20019920157735604796, 245733987135102838131
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2024

Keywords

Crossrefs

Cf. A002212, A270386, A371483.
Cf. A118971, A349332.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(5*k+3, k)/(k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+3,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^4 ).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349332.

A371523 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3 / (1-x))^2.

Original entry on oeis.org

1, 2, 15, 142, 1533, 17924, 220936, 2827218, 37202580, 500228562, 6842899886, 94931338876, 1332438761910, 18887047322030, 269986427261981, 3887654399820062, 56337997080499605, 821021578186212094, 12024687038651388155, 176900548019426869808, 2612917215947948178941
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371379, A371519, A371521, A371522.
Cf. A270386, A371518.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+1, k)/(5*k+2));

Formula

a(n) = 2 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+1,k)/(5*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349333.

A371483 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 18, 124, 933, 7446, 61943, 531348, 4666425, 41751325, 379230711, 3487769871, 32414437521, 303950138604, 2872137458010, 27322233357964, 261446381792670, 2514851398148595, 24303030755342128, 235841264063844258, 2297278004837062317
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2024

Keywords

Crossrefs

Cf. A002212, A270386, A371486.
Cf. A006632, A349331.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+2,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^3 ).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349331.

A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 15, 82, 477, 2901, 18235, 117555, 773085, 5166478, 34987170, 239570655, 1655933060, 11538839130, 80971109712, 571702698185, 4058556404958, 28951715755830, 207424064434502, 1491898838023884, 10768487956456506, 77977009814421534, 566310026687320290
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A045868, A371517, A371520, A371521.
Cf. A270386, A307678.
Cf. A371483.

Programs

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

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+2,k)/(2*k+3) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+3,k)/(k+1).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A307678.

A371519 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 45, 470, 5375, 65231, 825225, 10764185, 143739440, 1955340360, 27001732972, 377530388235, 5333865386885, 76031188364860, 1092117166466660, 15792298241897649, 229704197116753825, 3358528175751886765, 49333470827844265285, 727680248026484478405
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371379, A371521, A371522, A371523.
Cf. A270386, A371483, A371486.
Cf. A130564.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+4, k)/(k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+4,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^5 ).
G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349333.

A371518 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1-x))^2.

Original entry on oeis.org

1, 2, 11, 72, 525, 4104, 33647, 285526, 2486809, 22103726, 199697284, 1828472914, 16929944932, 158246198836, 1491210732346, 14151603542612, 135130396860130, 1297381593071890, 12516650939119421, 121281286192026308, 1179769340479567499
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349331, A371483, A371517.
Cf. A270386, A371523.

Programs

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

Formula

a(n) = 2 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+1,k)/(3*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349331.

A371576 G.f. satisfies A(x) = ( 1 + x*A(x)^(3/2) * (1 + x) )^2.

Original entry on oeis.org

1, 2, 9, 44, 240, 1390, 8404, 52426, 334964, 2180928, 14418123, 96525656, 653077411, 4458529390, 30674865164, 212472058410, 1480446579602, 10369560147798, 72972217926122, 515674254743332, 3657933383804959, 26036659997517572, 185905008055923918
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2024

Keywords

Crossrefs

Column k=2 of A378323.
Cf. A001629, A052705, A371577, A371578.
Cf. A270386, A364475.

Programs

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

Formula

a(n) = 2 * Sum_{k=0..n} binomial(3*k+2,k) * binomial(k,n-k)/(3*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364475.

A371583 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) / (1 - x) )^2.

Original entry on oeis.org

1, 2, 13, 104, 940, 9166, 94044, 1000602, 10939780, 122161128, 1387361151, 15974899766, 186069556707, 2188416960148, 25953579753464, 310022550197360, 3726709235290628, 45047517497268968, 547217895030263028, 6676784544374859088, 81789906534091716353
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2024

Keywords

Links

Crossrefs

Cf. A045868, A270386, A371518, A371523.
Cf. A349332, A371486, A371520.
Cf. A371578, A371585.

Programs

  • PARI
    a(n, r=2, s=1, t=5, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

a(n) = 2 * Sum_{k=0..n} binomial(5*k+2,k) * binomial(n-1,n-k)/(5*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349332.

A370695 G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/4) / (1-x))^4.

Original entry on oeis.org

1, 4, 22, 128, 777, 4872, 31330, 205560, 1370868, 9266104, 63343006, 437183260, 3042337215, 21323543252, 150395596016, 1066637271424, 7602188660799, 54422262148632, 391146728466980, 2821396586367568, 20417766975784066, 148200184917042112
Offset: 0

Views

Author

Seiichi Manyama, Mar 27 2024

Keywords

Crossrefs

Cf. A045902, A062109, A371379, A371486, A371517.
Cf. A270386, A307678, A371516.

Programs

  • Maple
    A370695 := proc(n)
    4*add(binomial(n-1,n-k)*binomial(3*k+4,k)/(3*k+4),k=0..n) ;
end proc:
seq(A370695(n),n=0..80) ; #R. J. Mathar, Oct 24 2024
  • PARI
    a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(3*k+4, k)/(3*k+4));

Formula

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+4,k)/(3*k+4).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A307678.
a(n) ~ 9 * 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Mar 29 2024
D-finite with recurrence 2*(n+2)*(2*n+3)*a(n) +(-55*n^2-74*n-15)*a(n-1) +6*(37*n^2-46*n-4)*a(n-2) -(295*n-319)*(n-3)*a(n-3) +124*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Oct 24 2024
Showing 1-10 of 10 results.

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