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

A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).

Original entry on oeis.org

1, 1, 5, 31, 219, 1678, 13570, 114014, 985542, 8708099, 78298727, 714105907, 6590200215, 61427125994, 577456943614, 5468604044500, 52122539760992, 499613409224137, 4813105582181533, 46576519080852235, 452545041339982871, 4413071971740021275, 43177663974461532959
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A002212, A002293, A307678, A317133, A346646 (partial sums), A349332, A349333, A349334, A349335.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^4/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1 + x A[x]^4/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]
  • 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)^4, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(4*k,k) / (3*k+1).
a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

Original entry on oeis.org

1, 1, 6, 46, 406, 3901, 39627, 418592, 4551672, 50610692, 572807157, 6577068383, 76426719408, 897078662538, 10620634999318, 126676885170703, 1520759193166329, 18361269213121164, 222814883564042704, 2716125963857227904, 33244557641365865109
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A002212, A307678, A349331, A349333, A349334, A349335.
Cf. A002294, A346647 (partial sums), A349361, A364748.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^5/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^5/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]
  • 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)^5, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
a(n) ~ 3381^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 15 2021
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = (4405*n^4 - 10346*n^3 + 9575*n^2 - 4354*n + 840)*a(n-1) - 12*(n-2)*(1255*n^3 - 3957*n^2 + 4492*n - 1820)*a(n-2) + 2*(n-3)*(n-2)*(10655*n^2 - 32733*n + 26908)*a(n-3) - 4*(n-4)*(n-3)*(n-2)*(3445*n - 6986)*a(n-4) + 3381*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Nov 17 2021

A349362 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^6 / (1 + x).

Original entry on oeis.org

1, 1, 5, 40, 370, 3740, 40006, 445231, 5102165, 59799505, 713496815, 8637432580, 105826926716, 1309793896431, 16351672606365, 205665994855320, 2603696877136060, 33151784577226295, 424258396639960591, 5454120586840761631, 70402732493668027775
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A001006, A002295, A127897, A317133, A346065 (binomial transform), A346666, A349333, A349361, A349363, A349364.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^6/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x]^6/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(6*k,k) / (5*k+1).
a(n) = (-1)^(n+1)* F([7/6, 4/3, 3/2, 5/3, 11/6, 1-n], [7/5, 8/5, 9/5, 2, 11/5], 6^6/5^5), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
a(n) ~ 43531^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021

A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

Original entry on oeis.org

1, 1, 9, 109, 1541, 23823, 390135, 6651051, 116798643, 2098313686, 38382509118, 712447023590, 13385500614902, 254065657922154, 4864482597112186, 93840443376075810, 1822169236520766546, 35586928273002974487, 698572561837366684479, 13775697096997873764647
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Comments

In general, for m > 1, Sum_{k=0..n} binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (1 + m^m/(m-1)^(m-1))^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 15 2021

Links

Crossrefs

Cf. A002212, A307678, A349331, A349332, A349333, A349334.
Cf. A007556, A346650 (partial sums), A349364.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^8/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^8/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
  • 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)^8, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).
a(n) ~ 17600759^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

Original entry on oeis.org

1, 1, 8, 85, 1051, 14197, 203064, 3022909, 46347534, 726894786, 11606936525, 188060979332, 3084087347910, 51094209834068, 853859480938095, 14376597494941454, 243649099741045190, 4153091242153905838, 71152973167920086796, 1224593757045581062444
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Links

Crossrefs

Cf. A002212, A307678, A349331, A349332, A349333, A349335.
Cf. A002296, A346649 (partial sums), A349363.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^7/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^7/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
  • 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)^7, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(7*k,k) / (6*k+1).
a(n) ~ 870199^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A371521 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^6.

Original entry on oeis.org

1, 6, 57, 614, 7158, 88002, 1123689, 14760024, 198172050, 2707560544, 37522666803, 526190125308, 7452866846847, 106465245105972, 1532129408941797, 22191180837313808, 323243244688652943, 4732225866305323686, 69591395772704207770, 1027547992261749954798
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371379, A371519, A371522, A371523.
Cf. A045868, A371516, A371517, A371520.

Programs

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

Formula

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

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.

A371379 G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/2) / (1-x))^4.

Original entry on oeis.org

1, 4, 34, 344, 3859, 46240, 579722, 7511272, 99782617, 1351784792, 18604380884, 259395030992, 3656180724752, 52011780756632, 745799171500502, 10768038899631476, 156414710103922340, 2284233700081510820, 33517461646190624690, 493917761019513208800
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371519, A371521, A371522, A371523.

Programs

  • PARI
    a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+3, k)/(5*k+4));

Formula

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+3,k)/(5*k+4).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349333.

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.

A371522 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1-x))^3.

Original entry on oeis.org

1, 3, 24, 235, 2586, 30603, 380359, 4896753, 64731747, 873539236, 11984536632, 166661420814, 2343950447112, 33282048811530, 476462982915993, 6869620848003570, 99663539644072305, 1453861111238442363, 21312207036239313936, 313783619269186619589
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A349333, A371379, A371519, A371521, A371523.

Programs

  • PARI
    a(n) = 3*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+2, k)/(5*k+3));

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+2,k)/(5*k+3).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349333.
Showing 1-10 of 12 results. Next

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

Content is available under The OEIS End-User License Agreement.