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 1-4 of 4 results.

A305540 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k different colors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 3, 1, 6, 6, 1, 10, 21, 12, 1, 14, 36, 24, 1, 22, 93, 132, 60, 1, 30, 150, 240, 120, 1, 46, 345, 900, 960, 360, 1, 62, 540, 1560, 1800, 720, 1, 94, 1173, 4980, 9300, 7920, 2520, 1, 126, 1806, 8400, 16800, 15120, 5040, 1, 190, 3801, 24612, 71400, 103320, 73080, 20160, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320
Offset: 1

Views

Author

Robert A. Russell, Jun 04 2018

Keywords

Comments

The number of achiral necklaces is equivalent to the number of achiral bracelets.

Examples

			The triangle begins with T(1,1):
1;
1,   1;
1,   2;
1,   4,     3;
1,   6,     6;
1,  10,    21,     12;
1,  14,    36,     24;
1,  22,    93,    132,     60;
1,  30,   150,    240,    120;
1,  46,   345,    900,    960,     360;
1,  62,   540,   1560,   1800,     720;
1,  94,  1173,   4980,   9300,    7920,    2520;
1, 126,  1806,   8400,  16800,   15120,    5040;
1, 190,  3801,  24612,  71400,  103320,   73080,   20160;
1, 254,  5796,  40824, 126000,  191520,  141120,   40320;
1, 382, 11973, 113652, 480060, 1048320, 1234800,  745920, 181440;
1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;
For a(4,2)=4, the achiral loops are AAAB, AABB, ABAB, and ABBB.

Crossrefs

Odd rows are A019538.
Even rows are A172106.
Columns 1-6 are A057427, A027383, A056489, A056490, A056491, and A056492.

Programs

  • Mathematica
    Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] + StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 15}, {k, 1, Ceiling[(n + 1)/2]}] // Flatten
  • PARI
    T(n, k) = (k!/2)*(stirling(floor((n+1)/2), k, 2)+stirling(ceil((n+1)/2), k, 2));
tabf(nn) = for(n=1, nn, for (k=1, ceil((n+1)/2), print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 02 2018

Formula

T(n,k) = (k!/2) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), where S2(n,k) is the Stirling subset number A008277.
T(n,k) = 2*A273891(n,k) - A087854(n,k).
G.f. for column k>1: (k!/2) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2). - Robert A. Russell, Sep 26 2018

A172107 Triangle T_3(n, m), the number of surjective multi-valued functions from {1, 1, 1, 2, 3, ..., n-2} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 1, 6, 9, 4, 1, 14, 45, 52, 20, 1, 30, 177, 388, 360, 120, 1, 62, 621, 2260, 3740, 2880, 840, 1, 126, 2049, 11524, 30000, 39720, 26040, 6720, 1, 254, 6525, 54292, 207620, 418320, 460320, 262080, 60480, 1, 510, 20337, 243268, 1309560, 3755640, 6150480, 5779200, 2903040, 604800
Offset: 1

Views

Author

Martin Griffiths, Jan 25 2010

Keywords

Comments

T_3(1, m) = T_3(2, m) = 0 by definition. T_3(n, m) also gives the number of ordered partitions of {1, 1, 1, 2, 3, ..., n-2} into exactly m parts.

Examples

			Triangle begins as:
  0;
  0,   0;
  1,   2,     1;
  1,   6,     9,      4;
  1,  14,    45,     52,      20;
  1,  30,   177,    388,     360,     120;
  1,  62,   621,   2260,    3740,    2880,     840;
  1, 126,  2049,  11524,   30000,   39720,   26040,    6720;
  1, 254,  6525,  54292,  207620,  418320,  460320,  262080,   60480;
  1, 510, 20337, 243268, 1309560, 3755640, 6150480, 5779200, 2903040, 604800;

Links

Crossrefs

This is related to A019538, A172106 and A172108.
Row sums give A172110.

Programs

  • Magma
    T:= func< n,k,m | n lt 3 select 0 else (&+[(-1)^(k+j)*Binomial(k,j)*Binomial(j+m-1,m)*j^(n-m): j in [1..k]]) >;
[T(n,k,3): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 14 2022
  • Mathematica
    f[r_, n_, m_]:= Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l,m}]; For[n = 3, n <= 10, n++, Print[Table[f[3, n, m], {m, 1, n}]]]
  • SageMath
    def T(n,k,m):
    if (n<3): return 0
    else: return sum( (-1)^(k-j)*binomial(k,j)*binomial(j+m-1,m)*j^(n-m) for j in (1..k) )
flatten([[T(n,k,3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 14 2022

Formula

T_3(n, m) = Sum_{j=0..m} binomial(m, j)*binomial(j+2, 3)*(-1)^(m-j)*j^(n-3), for n >= 3, with T(1, 1) = T(2, 1) = T(2, 2) = 0.
Sum_{k=1..n} T_3(n, k) = A172110(n).
Sum_{k=1..n} (-1)^k*T_3(n, k) = 0. - G. C. Greubel, Apr 14 2022

A172108 Triangle T_4(n, m), the number of surjective multi-valued functions from {1, 1, 1, 1, 2, 3, ..., n-3} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 1, 8, 18, 16, 5, 1, 18, 78, 136, 105, 30, 1, 38, 288, 856, 1205, 810, 210, 1, 78, 978, 4576, 10305, 12090, 7140, 1680, 1, 158, 3168, 22216, 74405, 134370, 134610, 70560, 15120, 1, 318, 9978, 101536, 483105, 1252650, 1882860, 1641360, 771120, 151200
Offset: 1

Views

Author

Martin Griffiths, Jan 25 2010

Keywords

Comments

T_4(1, m) = T_4(2, m) = T_4(3, m) = 0 by definition. T_4(n, m) also gives the number of ordered partitions of {1, 1, 1, 1, 2, 3, ..., n-3} into exactly m parts.

Examples

			Triangle begins as:
  0;
  0,   0;
  0,   0,    0;
  1,   3,    3,      1;
  1,   8,   18,     16,      5;
  1,  18,   78,    136,    105,      30;
  1,  38,  288,    856,   1205,     810,     210;
  1,  78,  978,   4576,  10305,   12090,    7140,    1680;
  1, 158, 3168,  22216,  74405,  134370,  134610,   70560,  15120;
  1, 318, 9978, 101536, 483105, 1252650, 1882860, 1641360, 771120, 151200;

Links

Crossrefs

This is related to A019538, A172106 and A172107.
Row sums give A172111.

Programs

  • Magma
    T:= func< n,k,m | n lt 4 select 0 else (&+[(-1)^(k+j)*Binomial(k,j)*Binomial(j+m-1,m)*j^(n-m): j in [1..k]]) >;
[T(n,k,4): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 14 2022
  • Mathematica
    f[r_, n_, m_]:= Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l,m}]; For[n = 4, n <= 10, n++, Print[Table[f[4, n, m], {m, 1, n}]]]
  • SageMath
    def T(n,k,m):
    if (n<4): return 0
    else: return sum( (-1)^(k-j)*binomial(k,j)*binomial(j+m-1,m)*j^(n-m) for j in (1..k) )
flatten([[T(n,k,4) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 14 2022

Formula

T_4(n, m) = Sum_{j=0..m} binomial(m,j)*binomial(j+3,4)*(-1)^(m-j)*j^(n-4), for n >= 4, with T(n, k) = 0 for n < 4.
Sum_{k=1.n} T_4(n, k) = A172111(n).
Sum_{k=1..n} (-1)^k*T_4(n, k) = 0. - G. C. Greubel, Apr 14 2022

A172109 a(n) is the number of ordered partitions of {1,1,2,3,...,n-1}.

Original entry on oeis.org

0, 2, 8, 44, 308, 2612, 25988, 296564, 3816548, 54667412, 862440068, 14857100084, 277474957988, 5584100659412, 120462266974148, 2772968936479604, 67843210855558628, 1757952715142990612, 48093560991292628228
Offset: 1

Views

Author

Martin Griffiths, Jan 25 2010

Keywords

Links

Crossrefs

Row sums of A172106.
Cf. A005649. - R. J. Mathar, Jan 28 2010
Cf. A083410.

Programs

  • Magma
    [(&+[Factorial(j+1)*StirlingSecond(n-1,j): j in [1..n]]): n in [1..30]]; // G. C. Greubel, Apr 14 2022
  • Mathematica
    f[r_, n_]:= Sum[Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l, m}], {m, n}]; Join[{0}, Table[f[2, n], {n,2,30}]]
  • PARI
    a(n) = sum(k=1, n-1, stirling(n-1,k,2)*(k+1)!); \\ Michel Marcus, Apr 14 2022
  • SageMath
    [sum( factorial(j+1)*stirling_number2(n-1,j) for j in (1..n-1) ) for n in (1..30)] # G. C. Greubel, Apr 14 2022

Formula

For n>=2, T_2(n) = Sum_{m=1..n} Sum_{l=0..m} C(m,l)*C(l+1,2)*(-1)^(m-l)*l^(n-2).
G.f.: 1/G(0) -1 where G(k) = 1 - x*(k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
G.f.: 1/Q(0) -1, where Q(k) = 1 - x*(3*k+2) - 2*x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) = Sum_{k=1..n-1} Stirling2(n-1,k)*(k+1)!. - Karol A. Penson, Sep 04 2015
a(n) ~ n! / (4 * log(2)^(n+1)). - Vaclav Kotesovec, Apr 15 2022
Showing 1-4 of 4 results.

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