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.

A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 31, 57, 113, 212, 410, 757, 1464, 2684, 5083, 9380, 17569, 32120, 59977, 109193, 202046, 367951, 675541, 1224453, 2243795, 4052369, 7377243, 13314989, 24140198, 43406515, 78510429, 140800279, 253663615, 454352111, 815790813, 1457485309
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 08 2018

Keywords

Links

Crossrefs

Cf. A000045, A023882, A075900, A077365, A179381, A318248.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1-Fibonacci[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

log(a(n)) ~ log(phi)*n + 2*sqrt(polylog(2, 1/sqrt(5))*n) - 3*(log(n)/4), where polylog(2, 1/sqrt(5)) = 0.5107013915606224266804289751265205446721... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio.

A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.

Original entry on oeis.org

1, 1, 2, 7, 19, 66, 212, 743, 2487, 9012, 31177, 113775, 404584, 1490726, 5376676, 20028981, 73068861, 273659672, 1009921813, 3801386137, 14125670266, 53477758556, 199950414035, 759566205693, 2857261603610, 10889590477287, 41136917417501, 157329747348492
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 22 2018

Keywords

Links

Crossrefs

Cf. A000108, A022629, A179381, A318248, A318263.

Programs

  • Maple
    C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1+CatalanNumber[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += CatalanNumber[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

Formula

a(n) ~ c * A000108(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + C(k)/4^k) = 2.608465265690846547082817204714986077801494... - Vaclav Kotesovec, Aug 24 2018

A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

Original entry on oeis.org

1, 4, 24, 112, 544, 2368, 10624, 44800, 190976, 791552, 3282944, 13414400, 54829056, 222117888, 899383296, 3625123840, 14601027584, 58659700736, 235555782656, 944552017920, 3786334535680, 15166305468416, 60736264994816, 243129089261568, 973133053952000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 09 2018

Keywords

Crossrefs

Cf. A075900, A300579.
Cf. A023882, A077365, A179381, A300520.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[1/(1-2^(k+1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 4^n, where c = A065446 = 1/QPochhammer(1/2) = 3.46274661945506361...

A179318 Triangle T(n,k) = A127742(n,k)/A048996(n,k) read by rows, k=1..A000041(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 14, 5, 2, 2, 1, 1, 1, 42, 14, 5, 4, 5, 2, 1, 2, 1, 1, 1, 132, 42, 14, 10, 14, 5, 4, 2, 5, 2, 1, 2, 1, 1, 1, 429, 132, 42, 28, 25, 42, 14, 10, 5, 4, 14, 5, 4, 2, 1, 5, 2, 1, 2, 1, 1, 1, 1430, 429, 132, 84, 70, 132, 42, 28, 25, 14, 10, 8, 42, 14, 10, 5, 4, 2, 14, 5, 4, 2, 1, 5, 2, 1, 2, 1
Offset: 1

Views

Author

Alford Arnold, Jul 12 2010

Keywords

Comments

The row lengths of A127742, A048996 and this triangle here are A000041(n).

Examples

			A127742 begins 1; 1, 1; 2, 2, 1; 5, 4, 1, 3, 1,
A048996 begins 1; 1, 1; 1, 2, 1; 1, 2, 1, 3, 1,
so
T(n,k) begins:
1
1 1
2 1 1
5 2 1 1 1
14 5 2 2 1 1 1

Crossrefs

Cf. A000041, A179381, A048996.

Formula

sum_{k=1..A000041(n)} T(n,k) = A179381(n).

Extensions

Edited and extended by R. J. Mathar, Jul 16 2010
A-numbers of row sums corrected by R. J. Mathar, Aug 01 2010

A306485 Expansion of Product_{k>=1} 1/(1 - Catalan(k)*x^k), where Catalan = A000108.

Original entry on oeis.org

1, 1, 3, 8, 26, 78, 271, 874, 3096, 10537, 37884, 132282, 484369, 1723568, 6362479, 23042165, 85706354, 313629597, 1175860079, 4340963778, 16355209663, 60882536222, 230370880224, 862533878347, 3278709952956, 12337333292318, 47042968508785, 177882993705004, 680221802560835, 2581438941995517
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 18 2019

Keywords

Links

Crossrefs

Cf. A000108, A088327, A179381.

Programs

  • Maple
    C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+C(i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019
  • Mathematica
    nmax = 29; CoefficientList[Series[Product[1/(1 - CatalanNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[Sum[CatalanNumber[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d CatalanNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} Catalan(j)^k*x^(j*k)/k).
a(n) ~ c * 4^n / (sqrt(Pi)*n^(3/2)), where c = Product_{k>=1} 1/(1 - Catalan(k) / 4^k) = 2.868839868502632... - Vaclav Kotesovec, Feb 23 2019
Showing 1-5 of 5 results.

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