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 21-26 of 26 results.

A246940 Number of partitions of n into 8 sorts of parts.

Original entry on oeis.org

1, 8, 72, 584, 4744, 38024, 304840, 2439368, 19520264, 156167944, 1249386824, 9995142472, 79961491848, 639692324232, 5117541421512, 40940334536648, 327522698972168, 2620181617189384, 20961453119350856, 167691625158581832, 1341533002724164744
Offset: 0

Views

Author

Alois P. Heinz, Sep 08 2014

Keywords

Links

Crossrefs

Column k=8 of A246935.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 8*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
  • Mathematica
    (O[x]^20 - 7/QPochhammer[8, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

G.f.: Product_{i>=1} 1/(1-8*x^i).
a(n) ~ c * 8^n, where c = Product_{k>=1} 1/(1-1/8^k) = 1.1635943971944701027405... . - Vaclav Kotesovec, Mar 19 2015
G.f.: Sum_{i>=0} 8^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A246941 Number of partitions of n into 9 sorts of parts.

Original entry on oeis.org

1, 9, 90, 819, 7461, 67239, 606051, 5455359, 49106502, 441967518, 3977783082, 35800130448, 322201861893, 2899817511237, 26098363809063, 234885281153616, 2113967586352095, 19025708339182545, 171231375557145825, 1541082380573345274, 13869741429702220707
Offset: 0

Views

Author

Alois P. Heinz, Sep 08 2014

Keywords

Links

Crossrefs

Column k=9 of A246935.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 9*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
  • Mathematica
    (O[x]^20 - 8/QPochhammer[9, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

G.f.: Product_{i>=1} 1/(1-9*x^i).
a(n) ~ c * 9^n, where c = Product_{k>=1} 1/(1-1/9^k) = 1.1408227572644372820166... . - Vaclav Kotesovec, Mar 19 2015
G.f.: Sum_{i>=0} 9^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Oct 18 2018

A246942 Number of partitions of n into 10 sorts of parts.

Original entry on oeis.org

1, 10, 110, 1110, 11210, 112210, 1123310, 11234310, 112355410, 1123567410, 11235798510, 112358120510, 1123582462610, 11235825994610, 112358272546710, 1123582739289710, 11235827519061810, 112358275329114810, 1123582754554197910, 11235827546928570910
Offset: 0

Views

Author

Alois P. Heinz, Sep 08 2014

Keywords

Links

Crossrefs

Column k=10 of A246935.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 10*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
  • Mathematica
    (O[x]^20 - 9/QPochhammer[10, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Formula

G.f.: Product_{i>=1} 1/(1-10*x^i).
a(n) ~ c * 10^n, where c = Product_{k>=1} 1/(1-1/10^k) = 1.12358275484865251117... . - Vaclav Kotesovec, Mar 19 2015

A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -2, 0, 0, 1, -4, -3, 2, 0, 0, 1, -5, -4, 6, 2, 1, 0, 1, -6, -5, 12, 6, 6, 0, 0, 1, -7, -6, 20, 12, 15, -2, 1, 0, 1, -8, -7, 30, 20, 28, -12, 2, 0, 0, 1, -9, -8, 42, 30, 45, -36, -3, -6, 0, 0, 1, -10, -9, 56, 42, 66, -80
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -2, -3, -4, ...
   0,  0,  2,  6, 12, ...
   0,  0,  2,  6, 12, ...

Links

Crossrefs

Columns k=0..3 give A000007, A010815, A070877, A292128.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292132.
Cf. A246935, A286957.

A292133 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, -1, 0, 1, -4, 6, -6, 1, 0, 1, -5, 12, -21, 14, -1, 0, 1, -6, 20, -52, 69, -26, 1, 0, 1, -7, 30, -105, 220, -201, 50, -1, 0, 1, -8, 42, -186, 545, -868, 591, -102, 2, 0, 1, -9, 56, -301, 1146, -2705, 3436, -1785, 214, -2, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,   1,   1, ...
   0, -1, -2,  -3,  -4, ...
   0,  0,  2,   6,  12, ...
   0, -1, -6, -21, -52, ...
   0,  1, 14,  69, 220, ...

Links

Crossrefs

Columns k=0..3 give A000007, A081362, A071109, A261582.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292134.
Cf. A246935, A286957.

A303914 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} 1/(1 - n*x^k).

Original entry on oeis.org

1, 2, 9, 55, 465, 5051, 69265, 1147287, 22307905, 497211049, 12484203601, 348391613615, 10691846920081, 357749800027465, 12958472141161457, 505088781523073326, 21076091000708067585, 937322034938743608556, 44256147057318887809993, 2210813717869831566759857, 116492226446226314836976401
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Links

Crossrefs

Cf. A000070, A124577, A246935, A303070.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> add(b(j$2, n), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 02 2018
  • Mathematica
    Table[SeriesCoefficient[1/(1 - x) Product[1/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - x) Exp[Sum[n^k x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] (1/(1 - x))*exp(Sum_{k>=1} n^k*x^k/(k*(1 - x^k))).
a(n) = Sum_{j=0..n} A246935(j,n).
a(n) ~ n^n. - Vaclav Kotesovec, May 04 2018
Previous Showing 21-26 of 26 results.

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