A005651 -id:A005651 - cp's OEIS frontend

A327803 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 3, 0, 7, 3, 0, 31, 16, 0, 121, 125, 0, 831, 711, 60, 0, 5041, 5915, 525, 0, 42911, 46264, 6328, 0, 364561, 438681, 67788, 0, 3742453, 4371085, 753420, 12600, 0, 39916801, 49321745, 8924685, 166320, 0, 486891175, 588219523, 113501784, 2966040

Offset: 0

Alois P. Heinz, Sep 25 2019

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       3;
  0,       7,       3;
  0,      31,      16;
  0,     121,     125;
  0,     831,     711,     60;
  0,    5041,    5915,    525;
  0,   42911,   46264,   6328;
  0,  364561,  438681,  67788;
  0, 3742453, 4371085, 753420, 12600;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Columns k=0-2 give: A000007, A061095, A327826.
Row sums give A005651.
Cf. A000217, A003056, A022915, A131632 (when the parts are distinct), A226874.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> nops({x[]})=k, partition(n))):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
      combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[x^Sign[j]*b[n - i*j, i-1]*multinomial[n, Join[{n-i*j}, Table[i, {j}]]], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 06 2020, after 2nd Maple program *)

T(n*(n+1)/2,n) = T(A000217(n),n) = A022915(n).

A104778 Table of values with shape sequence A000041 related to involutions and multinomials. Also column sums of the Kostka matrices associated with the partitions (in Abramowitz & Stegun ordering).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 4, 1, 2, 3, 5, 10, 1, 2, 3, 5, 7, 13, 26, 1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76, 1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232, 1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764, 1

Offset: 0

Alford Arnold, Mar 24 2005

Row sums give A178718.

			The 47 multinomials (corresponding to A005651(4)=47) can be distributed as in the following triangular array:
  1
  9 1
  4 6 1
  9 2 3 1
  1 3 2 3 1
divide each term by
  1
  3 1
  2 3 1
  3 2 3 1
  1 3 2 3 1
yielding
  1
  3 1
  2 2 1
  3 1 1 1
  1 1 1 1 1
with column sums 10 5 3 2 1.
Therefore the fourth row of the table is 1 2 3 5 10
The initial rows are:
  1,
  1,
  1, 2,
  1, 2, 4,
  1, 2, 3, 5, 10,
  1, 2, 3, 5, 7, 13, 26,
  1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76,
  1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232,
  1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764,
  ...

Wouter Meeussen, Table of n, a(n) for n = 0..372
Wouter Meeussen, Table of n, a(n), partition parts for n = 0..372

Cf. A000041, A000085, A005651, A036038, A097522, A104707, A104778, A178718.

A001475 and A000085 are subsequences.

(* for function 'kostka' see A178718 *)
aspartitions[n_] := Reverse /@ Sort[Sort /@ Partitions[n]];
asorder[n_] := rankpartition /@ Reverse /@ Sort[Sort /@ Partitions[n]];
Flatten[Table[Tr/@ Transpose[PadLeft[#,PartitionsP[k]] [[asorder[k]] ]&/@ kostka/@ aspartitions[k]],{k,11}]]

Corrected and edited by Wouter Meeussen, Jan 15 2012

A182926 Row sums of absolute values of A182928.

Original entry on oeis.org

1, 2, 3, 10, 25, 161, 721, 5706, 40881, 385687, 3628801, 41268613, 479001601, 6324319717, 87212177053, 1317906346186, 20922789888001, 357099708702023, 6402373705728001, 121882752536893635, 2432928081076384321, 51140835669924352717

Offset: 1

Peter Luschny, Apr 16 2011

nonn

The sum of multinomial coefficients can be computed recursively as
A005651(0) = 1 and A005651(n) = Sum_{1<=k<=n} binomial(n-1,k-1) * A182926(k) * A005651(n-k).
Möbius inversion yields: 1, 1, 2, 8, 24, 157, 720, 5696, 40878,...
A182927(2*i+1) = A182926(2*i+1).

			a(6) = 1 + 10 + 30 + 120 = 161.

Seiichi Manyama, Table of n, a(n) for n = 1..450

Cf. A182927, A182928, A005651, A007837.

A182926 := proc(n) local d;
add(n!/(d*((n/d)!)^d),d = numtheory[divisors](n)) end:
seq(A182926(i), i = 1..22);

a[n_] := Sum[ Abs[ -n!/(d*(-(n/d)!)^d)], {d, Divisors[n]}]; Table[ a[n], {n, 1, 22}] (* Jean-François Alcover, Jul 29 2013 *)

a(n) = Sum_{d|n} n!/(d*((n/d)!)^d).

E.g.f.: Sum_{k>=1} log(1/(1 - x^k/k!)). - Ilya Gutkovskiy, May 21 2019

A182927 Row sums of A182928.

Original entry on oeis.org

1, 0, 3, -8, 25, -99, 721, -5704, 40881, -340325, 3628801, -41245511, 479001601, -6129725315, 87212177053, -1317906346184, 20922789888001, -354320889234597, 6402373705728001, -121882630320799633, 2432928081076384321, -51041048673495232715

Offset: 1

Peter Luschny, Apr 16 2011

sign

The number of partitions of an n-set with distinct block sizes can
be computed recursively as A007837(0) = 1 and A007837(n) = - Sum_{1<=k<=n} binomial(n-1,k-1) * A182927(k) * A007837(n-k).
Möbius inversion yields: 1, -1, 2, -8, 24, -101, 720, -5696, 40878,...
A182927(2*i+1) = A182926(2*i+1)

			a(6) = 1 - 10 + 30 - 120 = -99.

Cf. A182926, A182928, A005651, A007837.

A182927 := proc(n) local d;
add(-n! / (d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:
seq(A182927(i), i = 1..22);

a[n_] := Sum[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[a[n], {n, 1, 22}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

a(n) = Sum_{d|n} -n!/(d*(-(n/d)!)^d).

E.g.f.: Sum_{k>=1} log(1 + x^k/k!). - Ilya Gutkovskiy, May 21 2019

A183610 Rectangular table where T(n,k) is the sum of the n-th powers of the k-th row of multinomial coefficients in triangle A036038 for n>=0, k>=0, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 5, 1, 1, 9, 46, 47, 7, 1, 1, 17, 244, 773, 246, 11, 1, 1, 33, 1378, 15833, 19426, 1602, 15, 1, 1, 65, 8020, 354065, 1980126, 708062, 11481, 22, 1, 1, 129, 47386, 8220257, 221300626, 428447592, 34740805, 95503, 30

Offset: 0

Paul D. Hanna, Aug 11 2012

			The table begins:
n=0: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...];
n=1: [1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, ...];
n=2: [1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, ...];
n=3: [1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, ...];
n=4: [1, 1, 17, 1378, 354065, 221300626, 286871431922, ...];
n=5: [1, 1, 33, 8020, 8220257, 25688403126, 199758931567152, ...];
n=6: [1, 1, 65, 47386, 194139713, 3033434015626, 141528428949437282, ...];
n=7: [1, 1, 129, 282124, 4622599553, 361140600078126, ...];
n=8: [1, 1, 257, 1686178, 110507041025, 43166813000390626, ...];
n=9: [1, 1, 513, 10097380, 2646977660417, 5169878244001953126, ...];
n=10:[1, 1, 1025, 60525226, 63465359844353, 619778904740009765626, ...];
...
The sums of the n-th power of terms in row k of triangle A036038 begin:
T(n,1) = 1^n,
T(n,2) = 1^n + 2^n,
T(n,3) = 1^n + 3^n + 6^n,
T(n,4) = 1^n + 4^n + 6^n + 12^n + 24^n,
T(n,5) = 1^n + 5^n + 10^n + 20^n + 30^n + 60^n + 120^n,
T(n,6) = 1^n + 6^n + 15^n + 20^n + 30^n + 60^n + 90^n + 120^n + 180^n + 360^n + 720^n, ...
Note that row n=0 forms the partition numbers A000041.

Paul D. Hanna, Table of rows 0..32 as read by antidiagonals, with index n = 0..528.

Cf. A036038, A000041, A005651, A183240, A183235, A183236, A183237, A183238, A215910 (main diagonal).

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
A:= (n, k)-> k!^n*b(k$2, n):
seq(seq(A(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Sep 11 2019

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n-i, Min[n-i, i], k]/i!^k + b[n, i-1, k]];
A[n_, k_] := k!^n b[k, k, n];
Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{T(n,k)=k!^n*polcoeff(1/prod(m=1, k, 1-x^m/m!^n +x*O(x^k)), k)}
for(n=0,10,for(k=0,8,print1(T(n,k),", "));print(""))

G.f. of row n: Sum_{k>=0} T(n,k)*x^k/k!^n = Product_{j>=1} 1/(1 - x^j/j!^n).

A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

Original entry on oeis.org

1, 1, 5, 244, 354065, 25688403126, 141528428949437282, 83257152559805973052807833, 7012360438832401192319979008881500417, 109324223115831487504443410090345278639832867784010, 396327911646787133737309113762487915762995734538047874429637296650

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

			The sums of the n-th power of multinomial coefficients in row n of triangle A036038 begin:
a(1) = 1^1 = 1;
a(2) = 1^2 + 2^2 = 5;
a(3) = 1^3 + 3^3 + 6^3 = 244;
a(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
a(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126;
a(6) = 1^6 + 6^6 + 15^6 + 20^6 + 30^6 + 60^6 + 90^6 + 120^6 + 180^6 + 360^6 + 720^6 = 141528428949437282;
a(7) = 1^7 + 7^7 + 21^7 + 35^7 + 42^7 + 105^7 + 140^7 + 210^7 + 210^7 + 420^7 + 630^7 + 840^7 + 1260^7 + 2520^7 + 5040^7 = 83257152559805973052807833; ...
which also form a logarithmic generating function of an integer series:
L(x) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...+ A215911(n)*x^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..30

Cf. A183610, A036038, A036740, A005651, A183240, A183235, A183236, A183237, A183238; A215911.

Cf. A326321.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
a:= n-> n!^n*b(n$3):
seq(a(n), n=0..12);  # Alois P. Heinz, Sep 11 2019

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n - i, Min[n - i, i], k]/i!^k + b[n, i - 1, k]];
a[n_] := n!^n b[n, n, n];
a /@ Range[0, 12] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

{a(n)=n!^n*polcoeff(1/prod(m=1, n, 1-x^m/m!^n +x*O(x^n)), n)}
for(n=1,15,print1(a(n),", "))

a(n) = [x^n/n!^n] * Product_{k=1..n} 1/(1 - x^k/k!^n) for n>=1, with a(0)=1.
Logarithmic derivative of A215911, ignoring the initial term a(0).
a(n) ~ (n!)^n = A036740(n). - Vaclav Kotesovec, Feb 19 2015
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

A320566 Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - x^k/k!).

Original entry on oeis.org

1, 2, 6, 23, 110, 617, 4035, 29927, 249926, 2316317, 23674841, 264329177, 3207278255, 42011308653, 591460307157, 8905905152798, 142897741683846, 2433947385964373, 43873382718719949, 834402502632550589, 16699964488044322205, 350869837371828862607, 7721899536993122262447

Offset: 0

Ilya Gutkovskiy, Oct 15 2018

nonn

Binomial transform of A005651.

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
N. J. A. Sloane, Transforms

Cf. A005651, A320567, A327870.

Row sums of A327801.

seq(coeff(series(factorial(n)*exp(x)*mul((1-x^k/factorial(k))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 22); # Muniru A Asiru, Oct 15 2018

nmax = 22; CoefficientList[Series[Exp[x] Product[1/(1 - x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x + Sum[Sum[x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]

E.g.f.: exp(x + Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)).
a(n) = Sum_{k=0..n} binomial(n,k)*A005651(k).
a(n) ~ exp(1) * A247551 * n!. - Vaclav Kotesovec, Jul 21 2019

A327711 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n (k is a partition length).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 27, 55, 171, 475, 1555, 4915, 20023, 68243, 288024, 1213828, 5435935, 23966970, 121432923, 578757824, 3130381590, 16427772974, 91877826663, 519546134163, 3199523135912, 18868494152257, 120274458082095, 772954621249540, 5219747666882153

Offset: 0

Alois P. Heinz, Sep 22 2019

nonn

Number of partitions of [n] whose block sizes are nondecreasing when blocks are ordered by their minima and these minima are {1..k} (for some k <= n). a(5) = 10: 12345, 13|245, 14|235, 15|234, 1|2345, 1|24|35, 1|25|34, 1|2|345, 1|2|3|45, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..635
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A005651, A179973, A326493, A327712, A327729.

with(combinat):
a:= n-> add(multinomial(n-nops(p), map(
    x-> x-1, p)[], 0), p=partition(n)):
seq(a(n), n=0..28);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<2, 0,
      b(n, i-1, p)) +b(n-i, min(n-i, i), p-1)/(i-1)!)
    end:
a:= n-> b(n$3):
seq(a(n), n=0..28);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 2, 0, b[n, i - 1, p]] + b[n - i, Min[n - i, i], p - 1]/(i - 1)!];
a[n_] := b[n, n, n];
a /@ Range[0, 28] (* Jean-François Alcover, May 01 2020, from 2nd Maple program *)

A076901 E.g.f.: 1/Product_{m>0} (1+(-x)^m/m!).

Original entry on oeis.org

1, 1, 1, 4, 21, 96, 520, 3795, 32053, 284368, 2763876, 30648465, 373339824, 4833294389, 67167087793, 1009753574739, 16215467043493, 275361718915824, 4947532173402532, 94054153646919213, 1882793796608183356, 39528099512321898363, 869222284280777733043

Offset: 0

Vladeta Jovovic, Nov 26 2002

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A005651, A182928.

A076901 := proc(n) local a, s;
s := proc(n) local d; add((-1)^d*n!/(d*(n/d)!^d),
d = numtheory[divisors](n)) end:
a := proc(n) option remember; local k;
`if`(n=0, 1, add(binomial(n-1,k-1)*s(k)*a(n-k),k = 1..n)) end:
(-1)^n*a(n) end:
seq(A076901(n), n=0..20);  # Peter Luschny, Apr 16 2011

nmax = 25; Table[SeriesCoefficient[Product[1/(1 + (-x)^k/k!), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 14 2017 *)

a(n) ~ c * n!, where c = Product_{k>=2} (1 + (-1)^k/k!) = 0.773515873861910082... - Vaclav Kotesovec, Sep 14 2017

A321752 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in p(u), where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, -2, 1, 0, 1, 3, -3, 1, 0, -2, 1, -4, 2, 4, -4, 1, 0, 0, 1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 5, -5, -5, 5, 5, -5, 1, 0, 0, 0, -2, 1, -6, 6, 6, 3, -2, -6, -12, 9, 6, -6, 1, 0, -4, 0, 2, 4, -4, 1, 0, 0, -6, 6, 3, -5, 1, 0, 0, 0, 0, 1, 7, -7, -7, -7, 14, 7, 7

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Triangle begins:
   1
   1
  -2   1
   0   1
   3  -3   1
   0  -2   1
  -4   2   4  -4   1
   0   0   1
   0   4   0  -4   1
   0   0   3  -3   1
   5  -5  -5   5   5  -5   1
   0   0   0  -2   1
  -6   6   6   3  -2  -6 -12   9   6  -6   1
   0  -4   0   2   4  -4   1
   0   0  -6   6   3  -5   1
   0   0   0   0   1
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   0   0   0   4   0  -4   1
For example, row 15 gives: p(32) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).

Wikipedia, Symmetric polynomial

Row sums are A321753.

Cf. A005651, A008480, A056239, A124794, A124795, A135278, A296150, A319193, A319225, A319226, A321742-A321765, A321854.

cp's OEIS Frontend

A327803 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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A104778 Table of values with shape sequence A000041 related to involutions and multinomials. Also column sums of the Kostka matrices associated with the partitions (in Abramowitz & Stegun ordering).

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A182926 Row sums of absolute values of A182928.

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A182927 Row sums of A182928.

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A183610 Rectangular table where T(n,k) is the sum of the n-th powers of the k-th row of multinomial coefficients in triangle A036038 for n>=0, k>=0, as read by antidiagonals.

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A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

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A320566 Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - x^k/k!).

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