A005651 -id:A005651 - cp's OEIS frontend

A072605 Number of necklaces with n beads over an n-ary alphabet {a1,a2,...,an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w.

1, 1, 2, 4, 13, 50, 270, 1641, 11945, 96784, 887982, 8939051, 99298354, 1195617443, 15619182139, 219049941201, 3293800835940, 52746930894774, 897802366250126, 16167544246362567, 307372573011579188, 6148811682561390279, 129164845357784003661

Offset: 0

Wouter Meeussen, Aug 06 2002

Alois P. Heinz, Table of n, a(n) for n = 0..200
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Eric Weisstein's world of Mathematics, Necklaces

Cf. A005651, A052307, A247551, A261531, A261599, A261600.

neck[li:{__Integer}] := Module[{n, d}, n=Plus@@li; d=n-First[li]; Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times@@((li/#2)!)&, 0, Divisors[GCD@@li]]/n]; Table[ Plus@@(neck /@ IntegerPartitions[n]), {n, 24}]

a(n)={if(n==0, 1, my(p=prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n))); sumdiv(n, d, eulerphi(n/d)*d!*polcoeff(p,d))/n)} \\ Andrew Howroyd, Dec 20 2017

a(n) = (1/n) * Sum_{d|n} phi(n/d) * A005651(d) for n > 0. - Andrew Howroyd, Sep 25 2017
See Mathematica line.
a(n) ~ c * (n-1)!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264818011615... . - Vaclav Kotesovec, Aug 27 2015

a(0)=1 prepended by Alois P. Heinz, Aug 23 2015

Name changed by Andrew Howroyd, Sep 25 2017

A185895 Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!)....

Original entry on oeis.org

1, -1, -1, 2, 3, 14, -40, -43, -357, -1762, 8004, 13067, 78540, 492439, 3932305, -26867293, -44643557, -363632466, -1729625764, -15939972937, -145669871232, 1488599170613, 3515325612655, 26765194180353, 151925998229148

Offset: 0

Michael Somos, Feb 05 2011

sign

From Peter Bala, Mar 17 2022: (Start)
Conjectures: 1) a(n) differs in sign from a(n-1) iff n is a triangular number (checked up to n = 1225 = (50*51)/2)
2) The same property holds for the coefficients of A(x)^2, the square of the o.g.f. A(x) =  1 - x - x^2 + 2*x^3 + 3*x^4 + ... : A(x)^2 = 1 - 2*x - x^2 + 6*x^3 + 3*x^4 + 18*x^5 - 110*x^6 - 22*x^7 - 483*x^8 - 2800*x^9 + 20030*x^10 + ....
3) The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..300

Cf. A005651, A007837, A168268.

{a(n) = if( n<0, 0, n! * polcoeff( prod( k=1, n, 1 - x^k / k!, 1 + x * O(x^n)), n))}

{a(n)=if(n<0,0,if(n==0,1,sum(k=1,n,(n-1)!/(n-k)!*a(n-k)*sumdiv(k,d,-d*d!^(-k/d)))))} [Hanna]

E.g.f.: Product_{k>0} (1 - x^k/k!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} -d*d!^(-k/d) and a(0) = 1 [cf. Vladeta Jovovic's formula in A007837].
E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

A321750 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 3, 6, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

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
   1   0
   1   2
   1   0   0
   1   1   0
   1   0   0   0   0
   1   3   6
   1   2   0   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
   1   2   2   2   0
   1   0   0   0   0   0   0   0   0   0   0
   1   1   0   0   0   0   0
   1   0   1   0   0   0   0
   1   6   4  12  24
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   1   1   2   2   0   0   0
For example, row 18 gives: p(221) = m(5) + 2m(32) + m(41) + 2m(221).

Wikipedia, Symmetric polynomial

Row sums are A321751.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A321742-A321765.

A183235 Sums of the cubes of multinomial coefficients.

Original entry on oeis.org

1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, 75263273895385, 54867365927680618, 54868847079435960134, 73030508546599681432983, 126197144644287414997433576, 277255161467330877411064074059

Offset: 0

Paul D. Hanna, Jan 04 2011

nonn

Equals sums of the cubes of terms in rows of the triangle of multinomial coefficients (A036038).

Ignoring initial term, equals the logarithmic derivative of A182963.

			G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 244*x^3/3!^3 + 15833*x^4/4!^3 +...
A(x) = 1/((1-x)*(1-x^2/2!^3)*(1-x^3/3!^3)*(1-x^4/4!^3)*...).
...
After the initial term a(0)=1, the next few terms are
a(1) = 1^3 = 1,
a(2) = 1^3 + 2^3 = 9,
a(3) = 1^3 + 3^3 + 6^3 = 244,
a(4) = 1^3 + 4^3 + 6^3 + 12^3 + 24^3 = 15833,
a(5) = 1^3 + 5^3 + 10^3 + 20^3 + 30^3 + 60^3 + 120^3 = 1980126, ...;
and continue with the sums of cubes of the terms in triangle A036038.

Vaclav Kotesovec, Table of n, a(n) for n = 0..180

Cf. A036038, A005651, A183240, A183236, A183237, A183238; A182963.

{a(n)=n!^3*polcoeff(1/prod(k=1, n, 1-x^k/k!^3 +x*O(x^n)), n)}

G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = Product_{n>=1} 1/(1 - x^n/n!^3).

a(n) ~ c * (n!)^3, where c = Product_{k>=2} 1/(1-1/(k!)^3) = 1.14825648754771664323845829539510031170864046029463094659207423270573478812675... . - Vaclav Kotesovec, Feb 19 2015

Examples added and name changed by Paul D. Hanna, Jan 05 2011

A325305 Irregular triangular array, read by rows: T(n,k) is the sum of the products of distinct multinomial coefficients (n_1 + n_2 + n_3 + ...)!/(n_1! * n_2! * n_3! * ...) taken k at a time, where (n_1, n_2, n_3, ...) runs over all integer partitions of n (n >= 0, 0 <= k <= A070289(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 1, 10, 27, 18, 1, 47, 718, 4416, 10656, 6912, 1, 246, 20545, 751800, 12911500, 100380000, 304200000, 216000000, 1, 1602, 929171, 260888070, 39883405500, 3492052425000, 177328940580000, 5153150631600000, 82577533320000000, 669410956800000000, 2224399449600000000, 1632586752000000000, 1, 11271

Offset: 0

Petros Hadjicostas, Sep 05 2019

This array was inspired by R. H. Hardin's recurrences for the columns of array A212855. Row n has length A070289(n) + 1.
This array differs from array A309951 starting at row n = 7. Array A309951 calculates a similar sum of products of multinomial coefficients, but the multinomial coefficients do not have to be distinct. Row n of array A309951 has length A000041(n) + 1, i.e., one more than the number of partitions of n.
Let P_n be the set of all lists a = (a_1, a_2,..., a_n) of integers a_i >= 0, i = 1,..., n such that 1*a_1 + 2*a_2 + ... + n*a_n = n; i.e., P_n is the set all integer partitions of n. (We use a different notation for partitions than the one in the name of T(n,k).) Then |P_n| = A000041(n) for n >= 0.
For n = 1..6, all the multinomial coefficients n!/((1!)^a_1 * (2!)^a_2 * ... * (n!)^a^n) corresponding to lists (a_1,...,a_n) in P_n are distinct; that is, A000041(n) = A070289(n) for n = 1..6.
For n = 7, the partitions (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (0, 2, 1, 0, 0, 0, 0) (i.e., 2 + 2 + 3) and  (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (3, 0, 0, 1, 0, 0, 0) (i.e., 1 + 1 + 1 + 4) give the same multinomial coefficient: 210 = 7!/(2!2!3!) = 7!/(1!1!1!4!). Hence, A000041(7) > A070289(7).
Looking at the multinomial coefficients of the integer partitions of n = 8, 9, 10 on pp. 831-832 of Abramowitz and Stegun (1964), we see that, even in these cases, we have A000041(n) > A070289(n).

			Triangle begins as follows:
  [n=0]: 1,   1;
  [n=1]: 1,   1;
  [n=2]: 1,   3,     2;
  [n=3]: 1,  10,    27,     18;
  [n=4]: 1,  47,   718,   4416,    10656,      6912;
  [n=5]: 1, 246, 20545, 751800, 12911500, 100380000, 304200000, 216000000;
  ...
For example, when n = 3, the integer partitions of 3 are 3, 1+2, 1+1+1, and the corresponding multinomial coefficients are 3!/3! = 1, 3!/(1!2!) = 3, and 3!/(1!1!1!) = 6. They are all distinct. Then T(n=3, k=0) = 1, T(n=3, k=1) = 1 + 3 + 6 = 10, T(n=3, k=2) = 1*3 + 1*6 + 3*6 = 27, and T(n=3, k=3) = 1*3*6 = 18.
Consider the list [1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040] of the A070289(7) = 15 - 1 = 14 distinct multinomial coefficients corresponding to the 15 integer partitions of 7. Then  T(7,0) = 1, T(7,1) = 11271 (sum of the coefficients), T(7,2) = 46169368 (sum of products of every two different coefficients), T(7,3) = 92088653622 (sum of products of every three different coefficients), and so on. Finally, T(7,14) = 2372695722072874920960000000000 = product of these coefficients.

Alois P. Heinz, Rows n = 0..15, flattened
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6), p. 248.
Wikipedia, Partition (number theory).

Cf. A000012 (column k=0), A000041, A005651, A070289, A212850, A212851, A212852, A212853, A212854, A212855, A212856, A309951, A309972, A325308 (column k=1).

g:= proc(n, i) option remember; `if`(n=0 or i=1, [n!], [{map(x->
      binomial(n, i)*x, g(n-i, min(n-i, i)))[], g(n, i-1)[]}[]])
    end:
b:= proc(n, m) option remember; `if`(n=0, 1,
      expand(b(n-1, m)*(g(m$2)[n]*x+1)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(nops(g(n$2)), n)):
seq(T(n), n=0..7);  # Alois P. Heinz, Sep 05 2019

g[n_, i_] := g[n, i] = If[n == 0 || i == 1, {n!}, Union[Map[Function[x, Binomial[n, i] x], g[n - i, Min[n - i, i]]], g[n, i - 1]]];
b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m] (g[m, m][[n]] x + 1)];
T[n_] := CoefficientList[b[Length[g[n, n]], n], x];
T /@ Range[0, 7] // Flatten (* Jean-François Alcover, May 06 2020, after Maple *)

Sum_{k=0..A070289(n)} (-1)^k * T(n,k) = 0.

A183236 Sums of multinomial coefficients to the 4th power.

Original entry on oeis.org

1, 1, 17, 1378, 354065, 221300626, 286871431922, 688780254549829, 2821284379712638737, 18510450092641988146882, 185104666826030540618018642, 2710117456989714966261367339909, 56196998736058707145628074314226034

Offset: 0

Paul D. Hanna, Jan 04 2011

nonn

Equals sums of the 4th power of terms in rows of the triangle of multinomial coefficients (A036038).

			G.f.: A(x) = 1 + x + 17*x^2/2!^4 + 1378*x^3/3!^4 + 354065*x^4/4!^4 +...
A(x) = 1/((1-x)*(1-x^2/2!^4)*(1-x^3/3!^4)*(1-x^4/4!^4)*...).

Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Cf. A036038, A005651, A183240, A183235, A183237, A183238.

{a(n)=n!^4*polcoeff(1/prod(k=1, n, 1-x^k/k!^4 +x*O(x^n)), n)}

G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = Product_{n>=1} 1/(1 - x^n/n!^4).

a(n) ~ c * (n!)^4, where c = Product_{k>=2} 1/(1-1/(k!)^4) = 1.067493570155257423039762074691753715853526744464586468822554194836450214299287... . - Vaclav Kotesovec, Feb 19 2015

A183237 Sums of multinomial coefficients to the 5th power.

Original entry on oeis.org

1, 1, 33, 8020, 8220257, 25688403126, 199758931567152, 3357348771315829641, 110013706232123658318433, 6496199364012472451887572970, 649619955166586474874295658148158, 104621943411970982740307507415589286391

Offset: 0

Paul D. Hanna, Jan 04 2011

nonn

Equals sums of the 5th power of terms in rows of the triangle of multinomial coefficients (A036038).

			G.f.: A(x) = 1 + x + 33*x^2/2!^5 + 8020*x^3/3!^5 + 8220257*x^4/4!^5 +...
A(x) = 1/((1-x)*(1-x^2/2!^5)*(1-x^3/3!^5)*(1-x^4/4!^5)*...).

Vaclav Kotesovec, Table of n, a(n) for n = 0..120

Cf. A036038, A005651, A183240, A183235, A183236, A183238.

{a(n)=n!^5*polcoeff(1/prod(k=1, n, 1-x^k/k!^5 +x*O(x^n)), n)}

G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = Product_{n>=1} 1/(1 - x^n/n!^5).

a(n) ~ c * (n!)^5, where c = Product_{k>=2} 1/(1-1/(k!)^5) = 1.03239096052278897179685563337623849923796538921602982416328969955606263213989... . - Vaclav Kotesovec, Feb 19 2015

A183238 Sums of multinomial coefficients to the 6th power.

Original entry on oeis.org

1, 1, 65, 47386, 194139713, 3033434015626, 141528428949437282, 16650678223240391821765, 4364875648285724481960633921, 2319673879587334552914376906604146, 2319673881714199597935597727665884813690

Offset: 0

Paul D. Hanna, Jan 04 2011

nonn

Equals sums of the 6th power of terms in rows of the triangle of multinomial coefficients (A036038).

			G.f.: A(x) = 1 + x + 65*x^2/2!^6 + 47386*x^3/3!^6 + 194139713*x^4/4!^6 +...
A(x) = 1/((1-x)*(1-x^2/2!^6)*(1-x^3/3!^6)*(1-x^4/4!^6)*...).

Cf. A036038, A005651, A183240, A183235, A183236, A183237.

{a(n)=n!^6*polcoeff(1/prod(k=1, n, 1-x^k/k!^6 +x*O(x^n)), n)}

G.f.: Sum_{n>=0} a(n)*x^n/n!^6 = Product_{n>=1} 1/(1 - x^n/n!^6).

A309973 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 2, 6, 10, 0, 2, 21, 42, 47, 0, 3, 42, 177, 264, 246, 0, 4, 90, 619, 1746, 2095, 1602, 0, 5, 176, 1809, 7556, 16085, 16608, 11481, 0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503, 0, 8, 640, 13961, 120964, 522890, 1262832, 1842659, 1503232, 871030

Offset: 0

Alois P. Heinz, Sep 21 2019

			T(3,1) = 2: 3aaa, 2aa1a.
T(3,2) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
T(3,3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   3;
  0, 2,   6,   10;
  0, 2,  21,   42,    47;
  0, 3,  42,  177,   264,    246;
  0, 4,  90,  619,  1746,   2095,   1602;
  0, 5, 176, 1809,  7556,  16085,  16608,  11481;
  0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-2 give: A000007, A000009 (for n>0), A327890.
Main diagonal gives A005651.
Row sums give A327679.
T(2n,n) gives A327681.
Cf. A327116, A327680.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
       binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *)

Sum_{k=1..n} k * T(n,k) = A327680(n).

A327801 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 47, 40, 18, 4, 1, 246, 235, 100, 30, 5, 1, 1602, 1476, 705, 200, 45, 6, 1, 11481, 11214, 5166, 1645, 350, 63, 7, 1, 95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1, 871030, 859527, 413316, 134568, 30996, 5922, 840, 108, 9, 1

Offset: 0

Alois P. Heinz, Sep 25 2019

Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.

			Triangle T(n,k) begins:
      1;
      1,     1;
      3,     2,     1;
     10,     9,     3,     1;
     47,    40,    18,     4,    1;
    246,   235,   100,    30,    5,   1;
   1602,  1476,   705,   200,   45,   6,  1;
  11481, 11214,  5166,  1645,  350,  63,  7, 1;
  95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Columns k=0-2 give: A005651, A327827, A327828.
Row sums give A320566.
T(2n,n) gives A266518.
T(n,n-1) gives A001477.
T(n+1,n-1) gives A045943.
Cf. A327869.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> k=0 or k in x, partition(n))):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
      `if`(i=k, 0, b(n-i, min(n-i, i), k)/i!))
    end:
T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i], k]/i!]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, from 2nd Maple program *)

cp's OEIS Frontend

A072605 Number of necklaces with n beads over an n-ary alphabet {a1,a2,...,an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w.

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A185895 Exponential generating function is (1-x^1/1!)(1-x^2/2!)(1-x^3/3!)....

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A321750 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

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A183235 Sums of the cubes of multinomial coefficients.

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PARI

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Extensions

A325305 Irregular triangular array, read by rows: T(n,k) is the sum of the products of distinct multinomial coefficients (n_1 + n_2 + n_3 + ...)!/(n_1! * n_2! * n_3! * ...) taken k at a time, where (n_1, n_2, n_3, ...) runs over all integer partitions of n (n >= 0, 0 <= k <= A070289(n)).

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Maple

Mathematica

Formula

A183236 Sums of multinomial coefficients to the 4th power.

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PARI

Formula

A183237 Sums of multinomial coefficients to the 5th power.

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PARI

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A183238 Sums of multinomial coefficients to the 6th power.

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