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 11-16 of 16 results.

A258461 Number of partitions of n into parts of exactly 6 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 22, 289, 2957, 26073, 208516, 1558219, 11087756, 76079368, 507834013, 3318628444, 21330627775, 135325210699, 849659799754, 5290544981423, 32722489513367, 201296535378562, 1232850239039750, 7523511821431264, 45777353199866275, 277862479920868778
Offset: 6

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

Column k=6 of A256130.
Cf. A320548.

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:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,6):
seq(a(n), n=6..30);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i](-1)^i/(i!(k - i)!), {i, 0, k}];
Table[T[n, 6], {n, 6, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 6^n, where c = 1/(6!*Product_{n>=1} (1-1/6^n)) = 1/(6!*QPochhammer[1/6, 1/6]) = 0.001723855087202395653855120059043... . - Vaclav Kotesovec, Jun 01 2015

A258462 Number of partitions of n into parts of exactly 7 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 29, 492, 6401, 70880, 704676, 6490951, 56524414, 471750267, 3810085912, 29989229859, 231255237311, 1754111872429, 13128442913712, 97189645384884, 713050007285941, 5192646586465458, 37581376345088462, 270593146237918806, 1939929376872664097
Offset: 7

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

Column k=7 of A256130.
Cf. A320549.

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:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,7):
seq(a(n), n=7..30);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
Table[T[n, 7], {n, 7, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 7^n, where c = 1/(7!*Product_{n>=1} (1-1/7^n)) = 1/(7!*QPochhammer[1/7, 1/7]) = 0.0002371101666331046535758625585353... . - Vaclav Kotesovec, Jun 01 2015

A258463 Number of partitions of n into parts of exactly 8 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 37, 788, 12705, 172520, 2084836, 23169639, 241881526, 2406802476, 23064505721, 214505275665, 1947297442670, 17332491414616, 151788374231505, 1311496639250495, 11205023121304298, 94832831557086797, 796244028801983324, 6640545376656071546
Offset: 8

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

Column k=8 of A256130.
Cf. A320550.

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:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,8):
seq(a(n), n=8..30);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
Table[T[n, 8], {n, 8, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 8^n, where c = 1/(8!*Product_{n>=1} (1-1/8^n)) = 1/(8!*QPochhammer[1/8, 1/8]) = 0.0000288589880256565005640019500910465339603... . - Vaclav Kotesovec, Jun 01 2015

A258464 Number of partitions of n into parts of exactly 9 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 46, 1202, 23523, 384227, 5542879, 73055550, 899381476, 10501235760, 117575627562, 1272685923724, 13401470756233, 137945728220761, 1393299928219604, 13851195993228228, 135865787060383171, 1317624915100561406, 12654868264707446322, 120534359759023523561
Offset: 9

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

Column k=9 of A256130.
Cf. A320551.

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:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,9):
seq(a(n), n=9..30);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
Table[T[n, 9], {n, 9, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 9^n, where c = 1/(9!*Product_{n>=1} (1-1/9^n)) = 1/(9!*QPochhammer[1/9, 1/9]) = 0.0000031438016899923866898607402658778352... . - Vaclav Kotesovec, Jun 01 2015

A258465 Number of partitions of n into parts of exactly 10 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 56, 1762, 41143, 795657, 13499449, 208050040, 2979881876, 40300054520, 520576172762, 6478447651345, 78185947269684, 919805200917658, 10591351937396242, 119764715367192468, 1333512940732309728, 14652754322423701707, 159182411488944508232
Offset: 10

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Comments

In general, column k>1 of A256130 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - Vaclav Kotesovec, Jun 01 2015

Links

Crossrefs

Column k=10 of A256130.
Cf. A320552.

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:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,10):
seq(a(n), n=10..30);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}];
Table[T[n, 10], {n, 10, 30}] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 10^n, where c = 1/(10!*Product_{n>=1} (1-1/10^n)) = 1/(10!*QPochhammer[1/10, 1/10]) = 0.0000003096292864992979803727261336621564... . - Vaclav Kotesovec, Jun 01 2015

A258457 Number of partitions of n into parts of exactly 2 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 4, 12, 30, 72, 160, 351, 743, 1561, 3219, 6616, 13456, 27312, 55139, 111166, 223472, 448902, 900305, 1804838, 3615137, 7239325, 14490368, 29000050, 58025059, 116090823, 232234573, 464554483, 929220024, 1858618215, 3717468189, 7435305664, 14871092926
Offset: 2

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

Column k=2 of A256130.

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:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,2):
seq(a(n), n=2..35);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
a[n_] := T[n, 2];
a /@ Range[2, 35] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 2^n, where c = 1/Product_{n>=2} (1-1/2^n) = 1/(2*A048651) = 1.7313733097275318... . - Vaclav Kotesovec, Jun 01 2015
Previous Showing 11-16 of 16 results.

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