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 1-7 of 7 results.

A047796 a(n) = Sum_{k=0..n} Stirling1(n,k)^2.

Original entry on oeis.org

1, 1, 2, 14, 194, 4402, 147552, 6838764, 418389078, 32639603798, 3161107700156, 372023906062756, 52280302234036252, 8645773770675973804, 1661888635268695003484, 367390786215560629372920, 92552610850186107484661670, 26356304249588730696338349990
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000275, A047797, A342111.

Programs

  • GAP
    List([0..20], n-> Sum([0..n], k-> Stirling1(n,k)^2 )); # G. C. Greubel, Aug 07 2019
  • Magma
    [(&+[StirlingFirst(n,k)^2: k in [0..n]]): n in [0..10]]; // G. C. Greubel, Aug 07 2019
  • Maple
    seq(add(stirling1(n, k)^2, k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019
  • Mathematica
    Table[Sum[StirlingS1[n,k]^2,{k,0,n}],{n,0,20}] (* Emanuele Munarini, Jul 04 2011 *)
  • Maxima
    makelist(sum(stirling1(n,k)^2,k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 1)^2); \\ Michel Marcus, Mar 26 2016
  • Sage
    [sum(stirling_number1(n,k)^2 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

Original entry on oeis.org

1, 0, 1, 6, 61, 770, 12160, 228382, 4989621, 124262532, 3475892685, 107901412520, 3681266754660, 136918473752216, 5513911474915116, 239034083286873630, 11098790133822288645, 549539910028075555016, 28903562131933534643851, 1609321474965547356327246
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2021

Keywords

Links

Crossrefs

Cf. A008277, A014322, A047797, A342111.
Cf. A048993.

Programs

  • Magma
    [(&+[StirlingSecond(n, k)*StirlingSecond(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021
  • Mathematica
    Table[Sum[StirlingS2[n, k]*StirlingS2[n, n-k], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*stirling(n, n-k, 2)); \\ Michel Marcus, Feb 28 2021
  • Sage
    [sum( stirling_number2(n, k)*stirling_number2(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

Formula

From Vaclav Kotesovec, Feb 28 2021, updated May 25 2025: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773047417401791158400820254382768364448971420138767247...
c = 1/(2*Pi*sqrt((1 + LambertW(-2*exp(-2)))*(3 + LambertW(-2*exp(-2))))) = 0.12826577250734152801558828593238744179869387423941684693208180123477... (End)

A363454 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements and no block contains both odd and even elements.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 11, 28, 87, 266, 952, 3381, 13513, 53915, 237113, 1046732, 5016728, 24186664, 125121009, 652084528, 3615047527, 20211789423, 119384499720, 711572380960, 4455637803543, 28162688795697, 186152008588691, 1242276416218540, 8636436319397292
Offset: 0

Views

Author

Alois P. Heinz, Jun 02 2023

Keywords

Examples

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 1: 1|2.
a(3) = 1: 13|2.
a(4) = 2: 13|24, 1|2|3|4.
a(5) = 4: 135|24, 13|2|4|5, 15|2|3|4, 1|2|35|4.
a(6) = 11: 135|246, 13|24|5|6, 13|26|4|5, 13|2|46|5, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46, 1|2|3|4|5|6.
a(7) = 28: 1357|246, 135|24|6|7, 137|24|5|6, 13|24|57|6, 135|26|4|7, 135|2|46|7, 137|26|4|5, 13|26|4|57, 137|2|46|5, 13|2|46|57, 13|2|4|5|6|7, 157|24|3|6, 15|24|37|6, 17|24|35|6, 1|24|357|6, 157|26|3|4, 15|26|37|4, 157|2|3|46, 15|2|37|46, 15|2|3|4|6|7, 17|26|35|4, 1|26|357|4, 17|2|35|46, 1|2|357|46, 1|2|35|4|6|7, 17|2|3|4|5|6, 1|2|37|4|5|6, 1|2|3|4|57|6.

Links

Crossrefs

Bisection gives A047797 (even part).
Cf. A000110, A124419, A124425, A363451.

Programs

  • Maple
    a:= n-> (h-> add(Stirling2(h, k)*Stirling2(n-h, k), k=0..h))(iquo(n, 2)):
seq(a(n), n=0..40);
# second Maple program:
b:= proc(n, x, y) option remember; `if`(abs(x-y)>n, 0,
      `if`(n=0, 1, `if`(x>0, b(n-1, y, x)*x, 0)+b(n-1, y, x+1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);

Formula

a(n) = Sum_{k=0..floor(n/2)} Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k).
a(2n) = A047797(n).

A345041 a(n) = Sum_{k=0..n} Stirling2(n,k)^n.

Original entry on oeis.org

1, 1, 2, 29, 3699, 10625002, 607758784933, 868305359018619811, 72322260589630363186583012, 141134946941935843819745493472571577, 21506852953850913182859127590586670415329232127, 213131394708948856925732826175269041102801068792839463406106
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2021

Keywords

Links

Crossrefs

Cf. A000110, A008277, A047797, A048993, A167010, A345040.

Programs

  • Magma
    [(&+[StirlingSecond(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022
  • Mathematica
    Table[Sum[StirlingS2[n, k]^n, {k, 0, n}], {n, 0, 11}]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)^n) \\ Felix Fröhlich, Jun 06 2021
  • SageMath
    def A345041(n): return sum(stirling_number2(n,j)^n for j in (0..n))
[A345041(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

A345040 a(n) = Sum_{k=0..n} Stirling2(n,k)^k.

Original entry on oeis.org

1, 2, 3, 12, 268, 25853, 19339964, 68901690994, 1638901380861357, 363916628499805466764, 384738112277336112497203088, 4821999492342155731355029409443825, 448660704121129122524211570743600451959266, 270068948293205668896252888517768674319536620944042
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2021

Keywords

Crossrefs

Cf. A000110, A008277, A047797, A048993, A167008, A345041.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS2[n, k]^k, {k, 0, n}], {n, 0, 13}]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)^k) \\ Felix Fröhlich, Jun 06 2021

A382793 a(n) = Sum_{k=0..n} (-1)^k * (Stirling2(n,k) * k!)^2.

Original entry on oeis.org

1, -1, 3, -1, -525, 21599, -575757, -11712961, 4147828275, -478419026401, 27474795508083, 3849481231073279, -1772585499434165325, 366912253456842693599, -26525609280231515934477, -17189616925094873258825281, 10414911263566240831226298675, -3136992122810471155294591778401
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2025

Keywords

Crossrefs

Cf. A047797, A048144, A192552.

Programs

  • Mathematica
    Table[Sum[(-1)^k (StirlingS2[n, k] k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(2 - Exp[x] - Exp[y] + Exp[x + y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / (2 - exp(x) - exp(y) + exp(x + y)).

A110855 Table T(n,k), n >= 0, k >= 0, product M*M^(T) where M is the lower triangular matrix in A048993 (Stirling2 numbers) and M^(T) denotes the transpose matrix of M, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 8, 11, 8, 1, 0, 0, 1, 16, 28, 28, 16, 1, 0, 0, 1, 32, 71, 87, 71, 32, 1, 0, 0, 1, 64, 184, 266, 266, 184, 64, 1, 0, 0, 1, 128, 491, 823, 952, 823, 491, 128, 1, 0, 0, 1, 256, 1348, 2598, 3381, 3381
Offset: 0

Views

Author

Philippe Deléham, Sep 17 2005

Keywords

Examples

			Matrix M:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  0, 1, 0, 0, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, 0, 0, ...
  0, 1, 3, 1, 0, 0, 0, 0, ...
  0, 1, 7, 6, 1, 0, 0, 0, ...
  ...
Matrix M^(T):
  1, 0, 0, 0, 0,  0, ...
  0, 1, 1, 1, 1,  1, ...
  0, 0, 1, 3, 7, 15, ...
  0, 0, 0, 1, 6, 25, ...
  0, 0, 0, 0, 1, 10, ...
  0, 0, 0, 0, 0,  1, ...
  ...
Table begins:
  1, 0,   0,   0,   0,   0,   0,   0,   0, 0, 0, ...
  0, 1,   1,   1,   1,   1,   1,   1,   1, 1,
  0, 1,   2,   4,   8,  16,  32,  64, 128, ...
  0, 1,   4,  11,  28,  71, 184, 491, ...
  0, 1,   8,  28,  87, 266, 823, ...
  0, 1,  16,  71, 266, 952, ...
  0, 1,  32, 184, 823, ...
  0, 1,  64, 491, ...
  0, 1, 128, ...
  0, 1, ...
  0, ...

Crossrefs

Diagonal sums: 1, 0, 1, 2, 4, 10, 29, 90, 295, ... see A000995.
Main diagonal: 1, 1, 2, 11, 87, 952, 13513, ... see A047797.
Showing 1-7 of 7 results.

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

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