A131689 -id:A131689 - cp's OEIS frontend

A299041 Irregular table: T(n,k) equals the number of alignments of length k of n strings each of length 3.

1, 1, 12, 30, 20, 1, 60, 690, 2940, 5670, 5040, 1680, 1, 252, 8730, 103820, 581700, 1767360, 3087000, 3099600, 1663200, 369600, 1, 1020, 94890, 2615340, 32186070, 214628400, 859992000, 2189325600, 3628409400, 3903900000, 2630628000, 1009008000, 168168000, 1, 4092, 979530, 58061420, 1411122300

Offset: 1

Peter Bala, Feb 02 2018

An alignment of n strings of various lengths is a way of inserting blank characters into the n strings so that the resulting strings all have the same length. We don't allow insertion of a blank character into the same position in each of the n strings.
In this case, let s_1,...,s_n be n strings each of length 3 over an alphabet A. Let - be a gap symbol not in A and let A' = union of A and {-}. An alignment of the n strings is an n-tuple (s_1',...,s_n') of strings each of length >= 3 over the alphabet A' such that
(a) the strings s_i', 1 <= i <= n, have the same length. This common length is called the length of the alignment.
(b) deleting the gap symbols from s_i' yields the string s_i for 1 <= i <= n
(c) there is no value j such that all the strings s_i', 1 <= i <= n have a gap symbol at position j.
By writing the strings s_i' one under another we can consider an alignment of n strings as an n X L matrix, where L, the length of the alignment, ranges from a minimum value of 3 to a maximum value of 3*n. Each row of the matrix has 3 characters from the alphabet A and (L - 3) gap characters.
For example,
  s_1' = ABC------
  s_2' = ---DEF---
  s_3' = ------GHI
is an alignment (of maximum length L = 9) of three strings s_1 = ABC, s_2 = DEF and s_3 = GHI each of length 3.
For the number of alignments of length k of n strings of length 1 (resp. 2) see A131689 (resp. A122193).

			Table begins
n\k| 3   4     5       6      7      8        9      10
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
  1| 1
  2| 1  12    30      20
  3| 1  60   690    2940   5670    5040    1680
  4| 1 252  8730  103820 581700 1767360 3087000 3099600 ...
...
T(2,5) = 30: An alignment of length 5 will have two gap symbols on each line. There are C(5,2) = 10 ways of choosing the 2 positions to insert the gap symbols in the first string. The second string in the alignment must then have nongap symbols at these two positions leaving three positions in which to insert the remaining 1 nongap symbol, giving in total 10 x 3 = 30 possible alignments of 2 strings of 3 characters. Some examples are
  ABC--   ABC--   ABC--
  D--EF   -D-EF   --DEF
Row 2: Sum_{i = 3..n-1} C(i,3)^2 = C(n,4) + 12*C(n,5) + 30*C(n,6) + 20*C(n,7).
Row 3: Sum_{i = 3..n-1} C(i,3)^3 = C(n,4) + 60*C(n,5) + 690*C(n,6) + 2940*C(n,7) + 5670*C(n,8)+ 5040*C(n,9)+ 1680*C(n,10).
exp( Sum_{n >= 1} R(n,2)*x^n/n ) = (1 + x + 153*x^2 + 128793*x^3 + 319155321*x^4 + 1744213657689*x^5 + ....)^8
exp( Sum_{n >= 1} R(n,3)*x^n/n ) = (1 + x + 424*x^2 + 998584*x^3 + 6925040260*x^4 + 105920615923684*x^5 + ....)^27.

P. Bala, Notes on A299041
M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
J. Engbers and C. Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Row sums A062208. Cf. A014606, A019538, A078741, A087127, A122193, A131689, A174266.

Cf. A086020, A086021, A086022.

seq(seq(add( (-1)^(k-i) *binomial(k,i)*binomial(i,3)^n, i = 0..k ), k = 3..3*n), n = 1..6);

nmax = 6; T[n_, k_] := Sum[(-1)^(k-i) Binomial[k, i] Binomial[i, 3]^n, {i, 0, k}]; Table[T[n, k], {n, 1, nmax}, {k, 3, 3n}] // Flatten (* Jean-François Alcover, Feb 20 2018 *)

T(n,k) = Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*binomial(i,3)^n.
T(n,3) = 1; T(n,3*n) = (3*n)!/6^n = A014606(n)
T(n,k) = binomial(k,3)*( T(n-1,k) + 3*T(n-1,k-1) + 3*T(n-1,k-2) + T(n-1,k-3) ) for 3 <= k <= 3*n with boundary conditions T(n,3) = 1 for n >= 1 and T(n,k) = 0 if (k < 3) or (k > 3*n).
Double e.g.f.: exp(-x)*Sum_{n >= 0} exp(binomial(n,3)*y)*x^n/n! = 1 + (x^3/3!)*y + (x^3/3! + 12*x^4/4! + 30*x^5/5! + 20*x^6/6!)*y^2/2! + ....
n-th row polynomial R(n,x) = Sum_{i >= 3} binomial(i,3)^n*x^i/(1 + x)^(i+1) for n >= 1.
1/(1 - x)*R(n,x/(1 - x)) = Sum_{i >= 3} binomial(i,3)^n*x^i for n >= 1.
R(n,x) = x^3 o x^3 o ... o x^3 (n factors), where o is the black diamond product of power series defined in Dukes and White.
R(n,x) = coefficient of (z_1)^3*...*(z_n)^3 in the expansion of the rational function 1/(1 + x - x*(1 + z_1)*...*(1 + z_n)).
The polynomials Sum_{k = 3..3*n} T(n,k)*x^(k-3)*(1 - x)^(3*n-k) are the row polynomials of A174266.
Sum_{i = 3..n-1} binomial(i,3)^m = Sum_{k = 3..3*m} T(m,k)*binomial(n,k+1) for m >= 1. See Examples below.
x^3*R(n,-1 - x) = (-1)^n*(1 + x)^3*R(n,x).
R(n+1,x) = 1/3!*x^3*(d/dx)^3 ((1 + x)^3*R(n,x)) for n >= 1.
The zeros of R(n,x) belong to the interval [-1, 0].
Row sums R(n,1) = A062208(n); alternating row sums R(n,-1) = (-1)^n.
For k a nonzero integer, the power series A(k,x) := exp( Sum_{n >= 1} 1/k^3*R(n,k)*x^n/n ) appear to have integer coefficients. See the Example section.
Sum_{k = 3..3*n} T(n,k)*binomial(x,k) = ( binomial(x,3) )^n. Equivalently, Sum_{k = 3..3*n} (-1)^(n+k)*T(n,k)*binomial(x+k,k) = ( binomial(x+3,3) )^n. Cf. the Worpitzky-type identity Sum_{k = 1..n} A019538(n,k)* binomial(x,k) = x^n.
Sum_{k = 3..3*n} T(n,k)*binomial(x,k-3) = -binomial(x,3)^n + 3*binomial(x+1,3)^n - 3*binomial(x+2,3)^n + binomial(x+3,3)^n. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane.

A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 10, 3, 1, 0, 75, 74, 21, 4, 1, 0, 541, 730, 219, 36, 5, 1, 0, 4683, 9002, 3045, 484, 55, 6, 1, 0, 47293, 133210, 52923, 8676, 905, 78, 7, 1, 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1, 0, 7087261, 45375130, 26857659, 5227236, 544505, 39390, 2359, 136, 9, 1

Offset: 0

Peter Luschny, May 21 2021

The array rows are recursively generated by applying the Akiyama-Tanigawa algorithm to the powers (see the Python implementation below). In this way the array becomes the image of A004248 under the AT-transformation when applied to the columns of A004248. This makes the array closely linked to A371761, which is generated in the same way, but applied to the rows of A004248. - Peter Luschny, Apr 27 2024

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,      1;
[3] 0, 3,      2,       1;
[4] 0, 13,     10,      3,       1;
[5] 0, 75,     74,      21,      4,      1;
[6] 0, 541,    730,     219,     36,     5,     1;
[7] 0, 4683,   9002,    3045,    484,    55,    6,    1;
[8] 0, 47293,  133210,  52923,   8676,   905,   78,   7,   1;
[9] 0, 545835, 2299754, 1103781, 194404, 19855, 1518, 105, 8, 1;
.
Seen as an array A(n, k) = T(n + k, n):
[0] [1, 0,   0,    0,     0,       0,         0, ...  A000007
[1] [1, 1,   3,   13,    75,     541,      4683, ...  A000670
[2] [1, 2,  10,   74,   730,    9002,    133210, ...  A004123
[3] [1, 3,  21,  219,  3045,   52923,   1103781, ...  A032033
[4] [1, 4,  36,  484,  8676,  194404,   5227236, ...  A094417
[5] [1, 5,  55,  905, 19855,  544505,  17919055, ...  A094418
[6] [1, 6,  78, 1518, 39390, 1277646,  49729758, ...  A094419
[7] [1, 7, 105, 2359, 70665, 2646007, 118893705, ...  A238464

Variant of the array is A094416 (which has column 0 and row 0 missing).
The coefficients of the Fubini polynomials are A131689.
Cf. A094420 (main diagonal of array), A372346 (row sums), A004248, A371761.

F := proc(n) option remember; if n = 0 then return 1 fi:
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
seq(seq(subs(x = k, F(n - k)), k = 0..n), n = 0..10);

F[n_] := F[n] = If[n == 0, 1,
   Expand[Sum[Binomial[n, k]*F[n - k]*x, {k, 1, n}]]];
Table[Table[F[n - k] /. x -> k, {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 06 2024, after Peter Luschny *)

# Computes the triangle.
@cached_function
def F(n):
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
def Fval(n): return [F(n - k).substitute(x = k) for k in (0..n)]
for n in range(10): print(Fval(n))

# Computes the square array using the Akiyama-Tanigawa algorithm.
def ATFubini(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = (n + 1)**k  # Chancing this to R[k] = k**n generates A371761.
        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(8): print([n], ATFubini(n, 7))  # Peter Luschny, Apr 27 2024

T(n, k) = (n - k)! * [x^(n - k)] (1 / (1 + k * (1 - exp(x)))).

T(2*n, n) = A094420(n).

A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 24, 25, 26, 30, 32, 33, 34, 35, 36, 40, 42, 45, 47, 48, 50, 51, 52, 54, 55, 60, 64, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 86, 90, 94, 96, 99, 100, 102, 104, 105, 108, 110, 119, 120, 123, 125, 126, 127, 128, 130

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  18: {{1},{2},{2}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  25: {{1,2},{1,2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}

For prime indices of prime indices we have A320456.
For binary indices of binary indices we have A326754.
An opposite version is A371292, A371293.
The case with squarefree product of prime indices is A371448.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],normQ[Join@@bpe/@prix[#]]&]

A037962 a(n) = n(15n^3 + 30n^2 + 5n - 2)*(n+4)!/5760.

Original entry on oeis.org

0, 1, 62, 1806, 40824, 834120, 16435440, 322494480, 6411968640, 130456085760, 2731586457600, 59056027430400, 1320663933388800, 30575780537702400, 733062897120153600, 18198613875746304000

Offset: 0

N. J. A. Sloane

nonn

For n>=1, a(n) is equal to the number of surjections from {1,2,...,n+4} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

Identity (1.20) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 3.

G. C. Greubel, Table of n, a(n) for n = 0..350
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A000142, A001298, A019538, A131689.

[Factorial(n)*StirlingSecond(n+4,n): n in [0..30]]; // G. C. Greubel, Jun 20 2022

Table[(n+4)!n(15n^3+30n^2+5n-2)/5760,{n,0,20}] (* Harvey P. Dale, Nov 16 2020 *)
Table[n!*StirlingS2[n+4, n], {n,0,30}] (* G. C. Greubel, Jun 20 2022 *)

[factorial(n)*stirling_number2(n+4,n) for n in (0..30)] # G. C. Greubel, Jun 20 2022

From G. C. Greubel, Jun 20 2022: (Start)
a(n) = (-1)^n * Sum_{j=0..n} (-1)^j * binomial(n, j)*j^(n+4).
a(n) = n!*StirlingS2(n+4, n).
a(n) = A131689(n+4, n).
a(n) = A019538(n+4, n).
E.g.f.: x*(1 + 22*x + 58*x^2 + 24*x^3)/(1-x)^9. (End)

A037963 a(n) = n^2(n+1)(3n^2 + 7n - 2)*(n+5)!/11520.

Original entry on oeis.org

0, 1, 126, 5796, 186480, 5103000, 129230640, 3162075840, 76592355840, 1863435974400, 45950224320000, 1155068769254400, 29708792431718400, 783699448602470400, 21234672840116736000, 591499300737945600000

Offset: 0

N. J. A. Sloane

nonn

For n>=1, a(n) is equal to the number of surjections from {1,2,...,n+5} onto {1,2,...,n}. - Aleksandar M. Janjic and Milan Janjic, Feb 24 2007

Identity (1.21) in H. W. Gould, Combinatorial Identities, Morgantown, 1972; page 3.

G. C. Greubel, Table of n, a(n) for n = 0..350
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A000142, A019538, A112494, A131689.

[Factorial(n)*StirlingSecond(n+5,n): n in [0..30]]; // G. C. Greubel, Jun 20 2022

Table[n!*StirlingS2[n+5, n], {n,0,30}] (* G. C. Greubel, Jun 20 2022 *)

[factorial(n)*stirling_number2(n+5,n) for n in (0..30)] # G. C. Greubel, Jun 20 2022

From G. C. Greubel, Jun 20 2022: (Start)
a(n) = (-1)^n * Sum_{j=0..n} (-1)^j * binomial(n, j)*j^(n+5).
a(n) = n!*StirlingS2(n+5, n).
a(n) = A131689(n+5, n).
a(n) = A019538(n+5, n).
E.g.f.: x*(1 + 52*x + 328*x^2 + 444*x^3 + 120*x^4)/(1-x)^11. (End)

A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

Original entry on oeis.org

1, 1, 0, 5, 1, 0, 61, 28, 1, 0, 1385, 1011, 123, 1, 0, 50521, 50666, 11706, 506, 1, 0, 2702765, 3448901, 1212146, 118546, 2041, 1, 0, 199360981, 308869464, 147485535, 24226000, 1130235, 8184, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:
F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.
F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.
F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.
F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.
The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.
Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:
F_{m} :   F_{0}    F_{1}    F_{2}    F_{3}    F_{4}
x = -1:  A165326  A155585  A002105  A292609  A292607
x =  1:  A000012  A000142  A000680  A014606  A014608  ... (m*n)!/m!^n
x =  0:    --     A000012  A000364  A002115  A211212  ... m-alternating permutations of length m*n.
Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.

			Triangle starts:
[n\k][    0        1        2       3     4  5  6]
--------------------------------------------------
[0][      1]
[1][      1,       0]
[2][      5,       1,       0]
[3][     61,      28,       1,      0]
[4][   1385,    1011,     123,      1,    0]
[5][  50521,   50666,   11706,    506,    1, 0]
[6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]

G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.

F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.
First column: A000364. Row sums: A000680. Alternating row sums: A002105.
Cf. A181985, A241171.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292604_row := proc(n) if n = 0 then return [1] fi;
add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292604_row(n) od;

T[n_, k_] /; 1 <= k <= n := T[n, k] = k (2 k - 1) T[n - 1, k - 1] + k^2 T[n - 1, k]; T[, 1] = 1; T[, _] = 0;
F[2, 0][] = 1; F[2, n][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
Table[row[n], {n, 0, 7}] (* Jean-François Alcover, Jul 06 2019 *)

def A292604_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..6): print(A292604_row(n))

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.

A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

Original entry on oeis.org

1, 2, 3, 6, 7, 22, 23, 32, 33, 48, 49, 86, 87, 112, 113, 516, 517, 580, 581, 1110, 1111, 1136, 1137, 1604, 1605, 5206, 5207, 5232, 5233, 5700, 5701, 8212, 8213, 9236, 9237, 13332, 13333, 16386, 16387, 16450, 16451, 17474, 17475, 21570, 21571, 24576, 24577

Offset: 1

Gus Wiseman, Mar 28 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    3: {{},{1}}
    6: {{1},{2}}
    7: {{},{1},{2}}
   22: {{1},{2},{3}}
   23: {{},{1},{2},{3}}
   32: {{1,2}}
   33: {{},{1,2}}
   48: {{3},{1,2}}
   49: {{},{3},{1,2}}
   86: {{1},{2},{3},{4}}
   87: {{},{1},{2},{3},{4}}
  112: {{3},{1,2},{4}}
  113: {{},{3},{1,2},{4}}
  516: {{2},{1,3}}
  517: {{},{2},{1,3}}
  580: {{2},{4},{1,3}}
  581: {{},{2},{4},{1,3}}

Without the covering condition we have A371289.
Without squarefree product we have A371292.
Interchanging binary and prime indices gives A371448.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts ordered set partitions, allowing empty sets A000629.
A005117 lists squarefree numbers.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A131689 counts patterns by number of distinct parts.
A302521 lists MM-numbers of set partitions, with empties A302505.
A326701 lists BII-numbers of set partitions.
A368533 lists numbers with squarefree binary indices, prime indices A302478.
Cf. A000040, A001222, A255906, A326782, A371291, A371294, A371447, A371452.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SquareFreeQ[Times @@ bpe[#]]&&normQ[Join@@prix/@bpe[#]]&]

Intersection of A371292 and A371289.

A371448 Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 26, 30, 32, 33, 34, 40, 47, 48, 51, 52, 55, 60, 64, 66, 68, 80, 85, 86, 94, 96, 102, 104, 110, 120, 123, 127, 128, 132, 136, 141, 143, 160, 165, 170, 172, 187, 188, 192, 204, 205, 208, 215, 220, 221, 226, 240, 246

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have (1) squarefree product and (2) binary indices covering an initial interval.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}
  33: {{2},{1,3}}
  34: {{1},{1,2,3}}
  40: {{1},{1},{1},{1,2}}
  47: {{1,2,3,4}}
  48: {{1},{1},{1},{1},{2}}
  51: {{2},{1,2,3}}

An opposite version is A371293, A371292.
Without the squarefree condition we have A371447, see also A320456, A326754.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000], SquareFreeQ[Times@@prix[#]]&&normQ[Join@@bpe/@prix[#]]&]

Intersection of A302505 and A371447.

A105796 "Stirling-Bernoulli transform" of Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 13, 25, 541, 1561, 47293, 181945, 7087261, 34082521, 1622632573, 9363855865, 526858348381, 3547114323481, 230283190977853, 1771884893993785, 130370767029135901, 1128511554418948441, 92801587319328411133, 892562598748128067705, 81124824998504073881821

Offset: 0

Paul Barry, Apr 20 2005

Alois P. Heinz, Table of n, a(n) for n = 0..425

Cf. A050946.

a:= n-> -add((-1)^k*k!*Stirling2(n+1, k+1)*(<<0|1>, <2|1>>^k)[1, 2], k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, May 09 2018

CoefficientList[Series[E^x*(1-E^x)/((2-E^x)*(1-2*E^x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 26 2013 *)

E.g.f.: e^x*(1-e^x)/((2-e^x)*(1-2*e^x)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * S2(n,k) * A001045(k).
a(n) ~ n! * (2-(-1)^n)/(6*log(2)^(n+1)). - Vaclav Kotesovec, Sep 26 2013
a(n) = Sum_{k = 0..n} (-1)^(n-k)*A131689(n,k)*A001045(k). - Philippe Deléham, May 25 2015

A133068 Number of surjections from an n-element set to an eight-element set.

Original entry on oeis.org

40320, 1451520, 30240000, 479001600, 6411968640, 76592355840, 843184742400, 8734434508800, 86355926616960, 823172919528960, 7621934141203200, 68937160460313600, 611692004959217280, 5342844138794426880, 46061530905262118400, 392795626402384128000

Offset: 8

Mohamed Bouhamida, Dec 16 2007

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Column k=8 of A019538 and A131689.

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

[&+[(-1)^(8-k)*Binomial(8, k)*k^n: k in [1..n]]: n in [8..25]]; // Vincenzo Librandi, Oct 21 2017

CoefficientList[Series[40320*x^8/((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(7*x - 1)*(8*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)
Table[Sum[(-1)^(8 - k)*Binomial[8, k]*k^n, {k, 1, 8}], {n, 8, 20}] (* G. C. Greubel, Oct 21 2017 *)

x='x+O('x^50); Vec(40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1))) \\ G. C. Greubel, Oct 20 2017

a(n) = Sum_{k=1..8} ((-1)^(8-k)*binomial(8,k)*k^n).
a(n) = A049434(n) * 8!. - Max Alekseyev, Nov 13 2009
G.f.: 40320*x^8/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Oct 25 2012
E.g.f.: (exp(x) - 1)^8. - Ilya Gutkovskiy, Jun 19 2018

Edited by N. J. A. Sloane, Jul 12 2008 at the suggestion of R. J. Mathar

More terms from Max Alekseyev, Nov 13 2009

cp's OEIS Frontend

A299041 Irregular table: T(n,k) equals the number of alignments of length k of n strings each of length 3.

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A344499 T(n, k) = F(n - k, k), where F(n, x) is the Fubini polynomial. Triangle read by rows, T(n, k) for 0 <= k <= n.

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A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

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A037962 a(n) = n*(15*n^3 + 30*n^2 + 5*n - 2)*(n+4)!/5760.

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A037963 a(n) = n^2*(n+1)*(3*n^2 + 7*n - 2)*(n+5)!/11520.

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A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

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A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

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A037962 a(n) = n(15n^3 + 30n^2 + 5n - 2)*(n+4)!/5760.

A037963 a(n) = n^2(n+1)(3n^2 + 7n - 2)*(n+5)!/11520.