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.

Previous Showing 41-44 of 44 results.

A178833 Partial sums of "Half-Catalan numbers" A000992.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 25, 49, 96, 199, 413, 894, 1924, 4261, 9392, 21205, 47534, 108492, 246313, 568003, 1302431, 3024429, 6990985, 16343338, 38026783, 89322813, 208986625, 493184761, 1159317065, 2745547588, 6480141829, 15399987104, 36475269692, 86916706534, 206503331542
Offset: 1

Views

Author

Jonathan Vos Post, Jan 01 2011

Keywords

Comments

The subsequence of primes begins: 2, 3, 5, 199, 4261, 493184761.
The subsequence of perfect powers begins: 1, 8, 25, 49.

Examples

			A000992 starts with 1, 1, 1, 2, 3, ... giving partial sums 1, 2, 3, 5, 8 ...

Crossrefs

Cf. A000992, A001190, A093637, A124973.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=1, 1,
      add(b(j)*b(n-j), j=1..n/2))
    end:
a:= proc(n) option remember; `if`(n<1, 0, b(n)+a(n-1)) end:
seq(a(n), n=1..42);  # Alois P. Heinz, Nov 04 2024
  • PARI
    lista(nn) = for (k=1, nn, print1(vecsum(A000992_list(k)), ", ")); \\ Michel Marcus, Feb 16 2015

Formula

a(n) = Sum_{i=1..n} A000992(i).

A203507 G.f.: Product_{n>=0} 1/(1-a(n)*x^(n+1))^3 = Sum_{n>=0} a(n)*x^n.

Original entry on oeis.org

1, 3, 15, 82, 504, 3198, 21592, 147570, 1045221, 7464052, 54549804, 400487997, 2990765270, 22396990002, 169881957174, 1291189065086, 9910770901971, 76178174174205, 590312326353680, 4578346159792815, 35745960436892046, 279290158338688617
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2012

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 82*x^3 + 504*x^4 + 3198*x^5 + 21592*x^6 +...
where
A(x) = 1/((1-x)*(1-3*x^2)*(1-15*x^3)*(1-82*x^4)*(1-504*x^5)*...)^3.
Related expansion:
A(x)^(1/3) = 1 + x + 4*x^2 + 19*x^3 + 110*x^4 + 659*x^5 + 4355*x^6 +...

Crossrefs

Cf. A093637, A093638.

Programs

  • PARI
    {a(n) = polcoeff(prod(k=0, n-1, 1/(1-a(k)*x^(k+1)+x*O(x^n)))^3, n)}

A299021 G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x*Product_{n>=1} (1 + a(n)*x^n)/(1 - a(n)*x^n).

Original entry on oeis.org

1, 2, 6, 22, 86, 358, 1558, 6966, 31894, 148918, 705062, 3380054, 16381158, 80056550, 394266950, 1955139942, 9749771926, 48873487942, 246160229782, 1244801094742, 6318514387638, 32184084454166, 164425969781062, 842429440124854, 4327629345403078, 22283328480744070
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 18 2018

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 86*x^5 + ... = x * ((1 + x) * (1 + 2*x^2) * (1 + 6*x^3) * (1 + 22*x^4) * (1 + 86*x^5) * ...) / ((1 - x) * (1 - 2*x^2) * (1 - 6*x^3) * (1 - 22*x^4) * (1 - 86*x^5) * ...).

Crossrefs

Cf. A032305, A073075, A093637.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[x Product[(1 + a[k] x^k)/(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 26}]
a[n_] := a[n] = SeriesCoefficient[x Exp[Sum[Sum[(1 + (-1)^(k + 1)) a[j]^k x^(j k)/k, {j, 1, n - 1}], {k, 1, n - 1}]], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 26}]

Formula

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x*exp(Sum_{k>=1} Sum_{n>=1} (1 + (-1)^(k+1))*a(n)^k*x^(n*k)/k).

A300411 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - a(k)*x^a(k)).

Original entry on oeis.org

1, 1, 1, 4, 9, 14, 19, 24, 45, 75, 105, 135, 229, 359, 503, 647, 1047, 1591, 2272, 2972, 4696, 6996, 9844, 12894, 20064, 29538, 41204, 54407, 84457, 123723, 171757, 225939, 348643, 508693, 703815, 923529, 1423892, 2076942, 2870977, 3763380, 5778379, 8414332, 11621307
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

Crossrefs

Cf. A006906, A093637, A151945, A196545, A291698, A293806, A293807, A296387.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - a[k] x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 42}]

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} 1/(1 - a(n)*x^a(n)).
Previous Showing 41-44 of 44 results.

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

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