A005651 -id:A005651 - cp's OEIS frontend

A356409 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^(1/(1-x)).

1, 1, 5, 28, 203, 1756, 17802, 205010, 2644287, 37669096, 586855058, 9914829508, 180429770402, 3516313661706, 73029591042943, 1609531482261375, 37504691293842367, 920966310015565936, 23764054962685200642, 642681497080268685092, 18174504398294667649782

Offset: 0

Seiichi Manyama, Aug 05 2022

nonn

Cf. A005651, A356025, A356407, A356408.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k!)^(1/(1-x))))

a356407(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356407(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A356407(k) * binomial(n-1,k-1) * a(n-k).

A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 10, 4, 14, 84, 1136, 967, 3342, 12823, 101762, 1769580

Offset: 0

Gus Wiseman, Aug 03 2025

An anti-run is a sequence with no adjacent equal terms.

			The a(7) = 4 anti-runs are:
  (1,2,1,2,1,2,1)
  (1,2,1,2,1,3,1)
  (1,2,1,3,1,2,1)
  (1,3,1,2,1,2,1)

For any multiplicities we have A005649.
For weakly instead of strictly decreasing multiplicities we have A321688.
A003242 and A335452 count anti-runs, ranks A333489.
A005651 counts ordered set partitions with weakly decreasing sizes, strict A007837.
A032020 counts strict anti-run compositions.
A325534 counts separable multisets, ranks A335433.
A325535 counts inseparable multisets, ranks A335448.
A336103 counts normal separable multisets, inseparable A336102.
A386583 counts separable partitions by length, inseparable A386584.
A386585 counts partitions of separable type by length, sums A336106, ranks A335127.
A386586 counts partitions of inseparable type by length, sums A025065, ranks A335126.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A000670, A019472, A106351, A238130, A335125, A335516, A386580, A386581.

seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
Table[Sum[Length[seps[y]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,10}]

A035796 Words over signatures (derived from multisets and multinomials).

Original entry on oeis.org

1, 1, 2, 2, 3, 18, 4, 48, 6, 5, 36, 100, 144, 6, 200, 180, 600, 7, 450, 900, 294, 24, 300, 1800, 8, 882, 7200, 448, 1200, 1470, 4410, 9, 1568, 22050, 648, 7200, 3136, 1800, 9408, 10, 14700, 2592, 16200, 1960, 56448, 900, 29400, 6048, 22050, 18144

Offset: 1

Alford Arnold

nonn

A reordering of A049009(n)=A049009(p(n)): distribution of words by numeric partition where the partition sequence: p(n)=[1],[2],[1,1],[3],[2,1],[1,1,1],[4],[3,1],[2,2],[2,1,1],... (A036036) is encoded by prime factorization ([P1,P2,P3,...] with P1 >= P2 >= P3 >= ... is encoded as 2^P1 * 3^P2 * 5^P3 *...): ep(n)=2,4,6,8,12,30,16,24,36,60, ... (A036035(n)) and then sorted: s(m)=2,4,6,8,12,16,24,30,32,36,48,60,... (A025487(m)). Hence A035796(n) = A049009(s(m)).

			27 = a(5) + a(6) + a(9) since a8(4) = 3, a12(5) = 18, a30(8) = 6; 256 = a(7) + a(8) + a(11) + a(13) + a(22) = 4 + 48 + 36 + 144 + 24
27 = a(5) + a(6) + a(9) = A049009(4) + A049009(5) + A049009(6) = 3 + 18 + 6 since A036035(4) = 8 = A025487(4+1), A036035(5) = 12 = A025487(5+1), A036035(6) = 30 = A025487(8+1);...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000312, A025487, A019575, A001700, A005651, A036035, A036036, A025487, A049009.

\\ here P is A025487 as vector and C is A049009 by partition.
GenS(lim)={my(L=List(), S=[1]); forprime(p=2, oo, listput(L, S); my(pp=vector(logint(lim, p), i, p^i)); S=concat([k*pp[1..min(if(k>1, my(f=factor(k)[, 2]); f[#f], oo), logint(lim\k, p))] | k<-S]); if(!#S, return(Set(concat(L)))) )}
P(n)={my(lim=1, v=[1]); while(#vt==S[k], sig))!) * prod(k=1, #sig, sig[k]!))}
seq(n)={[C(factor(t)[,2]) | t<-P(n)]} \\ Andrew Howroyd, Oct 18 2020

a(n) = A049009(p) where p is such that A036035(p) = A025487(n). [Corrected by Andrew Howroyd and Sean A. Irvine, Oct 18 2020]

More terms and additional comments from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001

a(1)=1 inserted by Andrew Howroyd and Sean A. Irvine, Oct 18 2020

A098545 Row sums of A098546.

Original entry on oeis.org

1, 3, 7, 21, 51, 148, 365, 983, 2461, 6360, 15687, 39757, 97033, 240425, 582622, 1421273, 3409861, 8222920, 19565707, 46680362, 110309476, 260876036, 612293443, 1437616751, 3354111156, 7823501148, 18157700800, 42112132458

Offset: 1

Alford Arnold, Sep 14 2004

By using multisets (cf. A001700) and multinomials (cf. A005651); A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end (compare A090657).

			A098546 begins
1
1 2
1 3 3
1 4 6 6 4
so sequence begins 1 3 7 21 ...

Cf. A000041, A000312.

a(n) = Sum_{k=1..n} binomial(n, k)*A008284(n, k). - Vladeta Jovovic, Jul 24 2005

More terms from Vladeta Jovovic, Jul 24 2005

A182928 Triangular array read by rows: [T(n,k),k=1..tau(n)] = [-n!/(d*(-(n/d)!)^d), d|n].

Original entry on oeis.org

1, 1, -1, 1, 2, 1, -3, -6, 1, 24, 1, -10, 30, -120, 1, 720, 1, -35, -630, -5040, 1, 560, 40320, 1, -126, 22680, -362880, 1, 3628800, 1, -462, 11550, -92400, -1247400, -39916800, 1, 479001600, 1, -1716, 97297200, -6227020800

Offset: 1

Peter Luschny, Apr 13 2011

The number of terms in the n-th row is the number of divisors of n. The n-th row is (apart from sign) a subsequence of the column labeled "M_1" for n-1 in Abramowitz and Stegun, Handbook, p. 831.
Let s(n) be the sum of row n. 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)*s(k)*A007837(n-k).
Let t(n) be the sum of the absolute values of row n. 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)*t(k)*A005651(n-k).

			The array starts with
[1] 1,
[2] 1,  -1,
[3] 1,   2,
[4] 1,  -3,   -6,
[5] 1,  24,
[6] 1, -10,   30,  -120,
[7] 1, 720,
[8] 1, -35,  -630, -5040,
[9] 1, 560, 40320,

Cf. A076901, A132958, A132959, A132960, A132962.

A182928_row := proc(n) local d;
seq(-n!/(d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:

row[n_] := Table[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

A292308 E.g.f.: 1/Product_{k>=1} (1 + x^k/k!).

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

Offset: 0

Seiichi Manyama, Sep 14 2017

sign

Cf. A005651, A007837, A076901.

nmax = 20; 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) = (-1)^n * A076901(n).

a(n) ~ c * (-1)^n * n!, where c = Product_{k>=2} (1 + (-1)^k/k!) = 0.77351587386... - Vaclav Kotesovec, Sep 14 2017

A309972 Product of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n.

Original entry on oeis.org

1, 1, 2, 18, 6912, 216000000, 1632586752000000000, 498266101635303733401600000000000, 1140494258799407218656986754465090350453096448000000000000000

Offset: 0

Alois P. Heinz, Aug 25 2019

nonn

			a(3) = M(3;3) * M(3;2,1) * M(3;1,1,1) = 1 * 3 * 6 = 18.

Alois P. Heinz, Table of n, a(n) for n = 0..14
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Rightmost terms in rows of A309951.

Cf. A000041, A005651, A036038, A078760, A210237.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [n!], [map(t->
      binomial(n, i)*t, b(n-i, min(n-i, i)))[], b(n, i-1)[]])
    end:
a:= n-> mul(i, i=b(n$2)):
seq(a(n), n=0..9);  # Alois P. Heinz, Aug 25 2019

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {n!}, Join[Binomial[n, i] #& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
a[n_] := Times @@ b[n, n];
a /@ Range[0, 9] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

a(n) = Product_{k=1..A000041(n)} A036038(n,k).

a(n) = A309951(n,A000041(n)).

A318808 Number of Lyndon permutations of a multiset whose multiplicities are the prime indices of n > 1.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 2, 6, 0, 6, 0, 4, 2, 1, 0, 12, 3, 1, 14, 5, 0, 10, 0, 24, 3, 1, 5, 30, 0, 1, 3, 20, 0, 15, 0, 6, 30, 1, 0, 60, 8, 20, 4, 7, 0, 90, 7, 30, 4, 1, 0, 60, 0, 1, 51, 120, 9, 21, 0, 8, 5, 35, 0, 180, 0, 1, 70, 9, 14, 28, 0, 120

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
The Lyndon product of two or more finite sequences is defined to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product.
a(1) = 1 by convention.

			The a(30) = 10 Lyndon permutations of {1,1,1,2,2,3}:
  (111223)
  (111232)
  (111322)
  (112123)
  (112132)
  (112213)
  (112312)
  (113122)
  (113212)
  (121213)

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Wikipedia, Lyndon word

Cf. A000670, A005651, A008480, A019536, A056239, A060223, A112798, A296372, A298941, A305936, A318762, A318810.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Permutations[nrmptn[n]],LyndonQ]],{n,2,100}]

sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
count(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, moebius(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
a(n)={if(n==1, 1, count(sig(n)))} \\ Andrew Howroyd, Dec 08 2018

a(p) = 0 for prime p. - Andrew Howroyd, Dec 08 2018

A321754 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and h is homogeneous 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

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).
Up to sign, same as A321752.

			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) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

Wikipedia, Symmetric polynomial

Row sums are all equal to 1.

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

A321897 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 3, 1, 0, 1, 1, 6, 3, 8, 6, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 1, 24, 30, 20, 15, 20, 10, 1, 0, 0, 0, 1, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, 0, 6, 0, 3, 8, 6, 1, 0, 0, 2, 3, 2, 4, 1, 0, 0, 0, 0, 1, 720, 840, 504, 420, 630

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    1
    0    1
    2    3    1
    0    1    1
    6    3    8    6    1
    0    0    1
    0    1    0    2    1
    0    0    2    3    1
   24   30   20   15   20   10    1
    0    0    0    1    1
  120   90  144   40   15   90  120   45   40   15    1
    0    6    0    3    8    6    1
    0    0    2    3    2    4    1
    0    0    0    0    1
  720  840  504  420  630  504  210  280  105  210  420  105   70   21    1
    0    0    0    1    0    2    1
For example, row 14 gives: 12h(41) = 6p(41) + 3p(221) + 8p(311) + 6p(2111) + p(11111).

Wikipedia, Symmetric polynomial

Row sums are A321898.

Cf. A005651, A008480, A056239, A124794, A124795, A319193, A321742-A321765, A321896.

cp's OEIS Frontend

A356409 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^(1/(1-x)).

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A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

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Mathematica

A035796 Words over signatures (derived from multisets and multinomials).

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A098545 Row sums of A098546.

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A182928 Triangular array read by rows: [T(n,k),k=1..tau(n)] = [-n!/(d*(-(n/d)!)^d), d|n].

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A292308 E.g.f.: 1/Product_{k>=1} (1 + x^k/k!).

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A309972 Product of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n.

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A318808 Number of Lyndon permutations of a multiset whose multiplicities are the prime indices of n > 1.

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