A000919 -id:A000919 - cp's OEIS frontend

A133132 Number of surjections from an n-element set to a ten-element set.

3628800, 199584000, 6187104000, 142702560000, 2731586457600, 45950224320000, 703098107712000, 10009442963520000, 134672620008326400, 1732015476199008000, 21473732319740064000, 258323865658578720000

Offset: 10

Mohamed Bouhamida, Dec 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 10..1000
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Column k=10 of A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A049435, A135456.

[10^n-10*9^n+45*8^n-120*7^n+210*6^n-252*5^n+210*4^n-120*3^n+45*2^n-10: n in [10..30]]; // Vincenzo Librandi, Apr 11 2012

With[{nn=30},Drop[CoefficientList[Series[(Exp[x]-1)^10,{x,0,nn}],x] Range[0,nn]!,10]] (* Harvey P. Dale, Sep 01 2016 *)

sum(k=1,10,(-1)^(10-k)*binomial(10,k)*k^n)

a(n) = 10^n-10*9^n+45*8^n-120*7^n+210*6^n-252*5^n+210*4^n-120*3^n+45*2^n-10.
a(n) = A049435(n) * 10!. - Max Alekseyev, Nov 13 2009
G.f.: 3628800*x^10/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x)-1)^10. - Alois P. Heinz, May 17 2016

More terms from Max Alekseyev, Nov 13 2009

Formula corrected by Charles R Greathouse IV, Mar 07 2010

A133360 Number of surjections from an n-element set to a nine-element set.

Original entry on oeis.org

362880, 16329600, 419126400, 8083152000, 130456085760, 1863435974400, 24359586451200, 297846188640000, 3457819037312640, 38528927611574400, 415357755774998400, 4358654246117808000, 44733116259693227520

Offset: 9

Mohamed Bouhamida, Dec 21 2007

Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

Column k=9 of A131689.

Cf. A000918, A000919, A000920, A001117, A001118, A049447, A135456.

a(n) = Sum_{k=1..9} (-1)^(9-k)*binomial(9,k)*k^n.
a(n) = A049447(n) * 9!. - Max Alekseyev, Nov 12 2009
G.f.: -362880*x^9/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^9. - Ilya Gutkovskiy, Jun 19 2018

More terms from Max Alekseyev, Nov 12 2009

A052776 a(n) = 4!nStirling2(n-1,4).

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 1440, 10920, 67200, 367416, 1864800, 9003720, 41976000, 190788312, 850930080, 3741028200, 16264684800, 70093951608, 299953709280, 1276255985160, 5404640136000, 22796986414104, 95836788986400, 401733800351400, 1679813323056000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 733
Index entries for linear recurrences with constant coefficients, signature (20,-170,800,-2273,3980,-4180,2400,-576).

Cf. A000919, A052749.

spec := [S,{B=Set(Z,1 <= card),S=Prod(Z,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0},Table[4!*n*StirlingS2[n-1,4],{n,30}]] (* Harvey P. Dale, Dec 24 2014 *)

a(n)={if(n>=1, 4!*n*stirling(n-1, 4, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

E.g.f.: x*exp(x)^4-4*exp(x)^3*x+6*exp(x)^2*x-4*x*exp(x)+x.
Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (1200*n+840*n^2+240*n^3+576+24*n^4)*a(n)+(-1200*n-1300*n^2-450*n^3-50*n^4)*a(n+1)+(35*n^4+420*n+665*n^2+280*n^3)*a(n+2)+(-80*n-10*n^4-140*n^2-70*n^3)*a(n+3)+(n^4+6*n^3+11*n^2+6*n)*a(n+4), a(5)=120}.
From Andrew Howroyd, Aug 08 2020: (Start)
a(n) = n*A000919(n-1) for n > 1.
G.f.: 24*x^5*(5 - 40*x + 105*x^2 - 100*x^3 + 24*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x))^2.
E.g.f.: x*(exp(x) - 1)^4. (End)

Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001

A305624 Number of chiral pairs of rows of n colors with exactly 4 different colors.

Original entry on oeis.org

0, 0, 0, 12, 120, 780, 4188, 20400, 93120, 409140, 1748220, 7337232, 30386160, 124696740, 508250988, 2061566400, 8331954240, 33585590580, 135115594140, 542784981552, 2178107091600, 8733341736900, 34996103558988, 140172672276000, 561255446475360, 2246716252964820, 8991948337723260, 35983044114659472, 143977928423467440, 576048972752188260, 2304607666801990188, 9219666007300387200, 36882370043723748480

Offset: 1

Robert A. Russell, Jun 06 2018

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

			For a(4) = 12, the chiral pairs are the 4! = 24 permutations of ABCD, each paired with its reverse.

Fourth column of A305622.
A056455(n) is number of achiral rows of n colors with exactly 4 different colors.
Cf. A000919, A056311.

k=4; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]

a(n) = my(k=4); (k!/2) * (stirling(n,k,2) - stirling(ceil(n/2),k,2)); \\ Michel Marcus, Jun 07 2018

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=4 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A000919(n) - A056455(n)) / 2.
a(n) = A000919(n) - A056311(n) = A056311(n) - A056455(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=4 colors used.

A371568 Array read by ascending antidiagonals: A(n, k) is the number of paths of length k in Z^n from the origin to points such that x1+x2+...+xn = k with x1,...,xn > 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 6, 1, 0, 0, 6, 14, 1, 0, 0, 0, 36, 30, 1, 0, 0, 0, 24, 150, 62, 1, 0, 0, 0, 0, 240, 540, 126, 1, 0, 0, 0, 0, 120, 1560, 1806, 254, 1, 0, 0, 0, 0, 1800, 8400, 5796, 510, 1

Offset: 1

Shel Kaphan, Mar 28 2024

T(n, k) can also be seen as the number of ordered partitions of k items into n nonempty buckets.
T(n, n) = n!, which is readily seen because to go from the origin to a point in Z^n a distance n away, with at least one step taken in each dimension, the first step can be in any of n dimensions, the second step in any of n-1 dimensions, and so on.
This array is the image of Pascal's triangle A007318 under the Akiyama-Tanigawa transformation. See the Python program. - Peter Luschny, Apr 19 2024

			 n\k 1 2 3  4   5    6     7      8       9       10
  --------------------------------------------------
 1|  1 1 1  1   1    1     1      1       1        1
 2|  0 2 6 14  30   62   126    254     510     1022
 3|  0 0 6 36 150  540  1806   5796   18150    55980
 4|  0 0 0 24 240 1560  8400  40824  186480   818520
 5|  0 0 0  0 120 1800 16800 126000  834120  5103000
 6|  0 0 0  0   0  720 15120 191520 1905120 16435440
 7|  0 0 0  0   0    0  5040 141120 2328480 29635200
 8|  0 0 0  0   0    0     0  40320 1451520 30240000
 9|  0 0 0  0   0    0     0      0  362880 16329600
10|  0 0 0  0   0    0     0      0       0  3628800

Cf. A000918 (n=2), A001117 (n=3), A000919 (n=4), A001118 (n=5), A000920 (n=6).
Cf. A135456 (n=7), A133068 (n=8), A133360 (n=9), A133132 (n=10).
Cf. A371064, A007318.
See A019538 and A131689 for other versions.

A[n_,k_] := Sum[(-1)^(n-i) * i^k * Binomial[n,i], {i,1,n}]

# The Akiyama-Tanigawa algorithm for the binomial generates the rows.
# Adds row(0) = 0^k and column(0) = 0^n.
from math import comb as binomial
def ATBinomial(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = binomial(k, n)
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(11): print([n], ATBinomial(n, 11))  # Peter Luschny, Apr 19 2024

A(n,k) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * i^k

A056269 Number of primitive (aperiodic) words of length n which contain exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 24, 240, 1560, 8400, 40800, 186480, 818280, 3498000, 14674440, 60780720, 249393480, 1016542560, 4123132800, 16664094960, 67171179600, 270232006800, 1085569963080, 4356217672800, 17466683473800

Offset: 1

Marks R. Nester

nonn

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A000302, A000919, A054719.

sum mu(d)*A000919(n/d) where d|n.

Seems to be n * A056289.

cp's OEIS Frontend

A133132 Number of surjections from an n-element set to a ten-element set.

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A133360 Number of surjections from an n-element set to a nine-element set.

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A052776 a(n) = 4!*n*Stirling2(n-1,4).

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A305624 Number of chiral pairs of rows of n colors with exactly 4 different colors.

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A371568 Array read by ascending antidiagonals: A(n, k) is the number of paths of length k in Z^n from the origin to points such that x1+x2+...+xn = k with x1,...,xn > 0.

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A056269 Number of primitive (aperiodic) words of length n which contain exactly four different symbols.

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Formula

A052776 a(n) = 4!nStirling2(n-1,4).