A005651 -id:A005651 - cp's OEIS frontend

A321914 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

1, -2, 1, 1, 0, 3, -3, 1, -3, 1, 0, 1, 0, 0, -4, 2, 4, -4, 1, 2, 1, -2, 0, 0, 4, -2, -1, 1, 0, -4, 0, 1, 0, 0, 1, 0, 0, 0, 0, 5, -5, -5, 5, 5, -5, 1, -5, 1, 5, -3, -1, 1, 0, -5, 5, -1, 1, -2, 0, 0, 5, -3, 1, 0, 0, 0, 0, 5, -1, -2, 0, 1, 0, 0, -5, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 22 2018

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

Also the coefficient of h(v) in f(u), where f is forgotten symmetric functions and h is homogeneous symmetric functions.

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -2  1
  (11):  1
.
  (3):    3 -3  1
  (21):  -3  1
  (111):  1
.
  (4):    -4  2  4 -4  1
  (22):    2  1 -2
  (31):    4 -2 -1  1
  (211):  -4     1
  (1111):  1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):    -5  1  5 -3 -1  1
  (32):    -5  5 -1  1 -2
  (221):    5 -3  1
  (311):    5 -1 -2     1
  (2111):  -5  1
  (11111):  1
For example, row 14 gives: m(32) = -5e(5) - e(32) + 5e(41) + e(221) - 2e(311).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321746.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321854, A321738, A321912-A321935.

A321918 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in p(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, -2, 1, 0, 1, 3, -3, 1, 0, -2, 1, 0, 0, 1, -4, 2, 4, -4, 1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 1, 5, -5, -5, 5, 5, -5, 1, 0, -4, 0, 2, 4, -4, 1, 0, 0, -6, 6, 3, -5, 1, 0, 0, 0, 4, 0, -4, 1, 0, 0, 0, 0, 3, -3, 1, 0, 0, 0, 0, 0, -2, 1

Offset: 1

Gus Wiseman, Nov 22 2018

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

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -2  1
  (11):     1
.
  (3):    3 -3  1
  (21):     -2  1
  (111):        1
.
  (4):    -4  2  4 -4  1
  (22):       4    -4  1
  (31):          3 -3  1
  (211):           -2  1
  (1111):              1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):       -4     2  4 -4  1
  (32):          -6  6  3 -5  1
  (221):             4    -4  1
  (311):                3 -3  1
  (2111):                 -2  1
  (11111):                    1
For example, row 14 gives: p(32) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321752.

Cf. A005651, A008480, A056239, A124794, A124795, A135278, A215366, A318284, A319191, A319193, A319225, A319226, A321912-A321935.

A075197 Number of partitions of n balls of n colors.

Original entry on oeis.org

1, 1, 6, 38, 305, 2777, 28784, 330262, 4152852, 56601345, 829656124, 12992213830, 216182349617, 3804599096781, 70540645679070, 1373192662197632, 27982783451615363, 595355578447896291, 13193917702518844859, 303931339674133588444, 7263814501407389465610

Offset: 0

Christian G. Bower, Sep 07 2002

nonn

For each integer partition of n, consider each part of size k to be a box containing k balls of up to n color. Order among parts and especially among parts of the same size does not matter. - Olivier Gérard, Aug 26 2016

			Illustration of first terms, ordered by number of parts, size of parts and smallest color of parts, etc.
a(1) = 1:
  {{1}}
a(2) = 6 = 3+3:
  {{1,1}},{{1,2}},{{2,2}},
  {{1},{1}},{{1},{2}},{{2},{2}}
a(3) = 38 = 10+18+10:
  {{1,1,1}},{{1,1,2}},{{1,1,3}},{{1,2,2}},{{1,2,3}},{{1,3,3}},
  {{2,2,2}},{{2,2,3}},{{2,3,3}},{{3,3,3}},
  {{1},{1,1}},{{1},{1,2}},{{1},{1,3}},{{1},{2,2}},{{1},{2,3}},{{1},{3,3}},
  {{2},{1,1}},{{2},{1,2}},{{2},{1,3}},{{2},{2,2}},{{2},{2,3}},{{2},{3,3}},
  {{3},{1,1}},{{3},{1,2}},{{3},{1,3}},{{3},{2,2}},{{3},{2,3}},{{3},{3,3}},
  {{1},{1},{1}},{{1},{1},{2}},{{1},{1},{3}},{{1},{2},{2}},{{1},{2},{3}},{{1},{3},{3}},
  {{2},{2},{2}},{{2},{2},{3}},{{2},{3},{3}},{{3},{3},{3}}

Alois P. Heinz, Table of n, a(n) for n = 0..400

Main diagonal of A075196.
Cf. A001700 (n balls of one color in n unlabeled boxes).
Cf. A209668 (boxes are ordered by size but not by content among a given size: order among boxes of the same size matters.),
Cf. A261783 (compositions of balls of n colors: boxes are labeled)
Cf. A252654 (lists instead of boxes : order of balls matter)
Cf. A000262 (lists instead of boxes and all n colors are used)
Cf. A255906 (the c colors used form the interval [1,c])
Cf. A255951 (the n-1 colors used form the interval [1,n-1])
Cf. A255942 (0/1 binary coloring)
Cf. A066186 (only 1 color among n = n * p(n))
Cf. A000110 (the n possible colors are used : set partitions of [n])
Cf. A005651 (the n possible colors are used and order of parts of the same size matters)
Cf. A000670 (the n possible colors are used and order of all parts matters)

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-1, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 26 2012

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]/n]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = [x^n] Product_{k>=1} 1 / (1 - x^k)^binomial(k+n-1,n-1). - Ilya Gutkovskiy, May 09 2021

A076900 Expansion of e.g.f.: 1/Product_{m>0} (1-x^m/(m-1)!).

Original entry on oeis.org

1, 1, 4, 15, 88, 505, 4056, 31549, 311816, 3083049, 36343720, 431215741, 5937234348, 82236865165, 1291252453050, 20477737537755, 361495828272496, 6449450737736065, 126566562342343176, 2509520619696338269, 54179963857121953460, 1182248224137860933781

Offset: 0

Vladeta Jovovic, Nov 26 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A005651, A032299.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(n, i)*i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + If[i > n, 0, b[n-i, i] Binomial[n, i] i]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 03 2020, after Alois P. Heinz *)

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*((j - 1)!)^k)). - Ilya Gutkovskiy, Sep 13 2018

a(n) ~ c * n * n!, where c = A247551/2. - Vaclav Kotesovec, Sep 13 2018

A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 6, 120, 60, 20, 5, 1, 30, 10, 720, 360, 120, 30, 6, 1, 180, 60, 15, 20, 90

Offset: 1

Alford Arnold, Mar 27 2008

Multinomials count permutations of multisets and also paths in lattices; for example, there are six paths (from null to full) through the lattice of divisors for signature 36: 2233 2323 2332 3223 3232 and 3322.

			a(11) is six because the eleventh least prime signature in source format is 36 the signature for partition 2+2 the ninth partition and A036038(9) = 6.
The tables begin:
1.......2.......6.......24......120.....720....5040.....40320......362880
........1.......3.......12.......60.....360....2520.....20160......181440
................1.......4........20.....120.....840......6720.......60480
........................1........5.......30.....210......1680.......15120
.. ..............................1........6......42......336........3024
..........................................1.......7.......56.........504
..................................................1........8..........72
...........................................................1...........9
.......................................................................1
........................6........30.....180....1260....10080........90720
.................................10......60.....420.....3360........30240
...

Cf. A000142 A001710 A005651 (column sums) A025487 A053445.

Cf. A173333. [From Reinhard Zumkeller, Feb 19 2010]

A143463 Number of multiple hierarchies for n labeled elements.

Original entry on oeis.org

1, 4, 20, 133, 1047, 9754, 103203, 1229330, 16198452, 234110702, 3675679471, 62287376870, 1132138152251, 21963847972941, 452786198062541, 9881445268293457, 227522503290656371, 5510876754647261442, 140040543831299600637, 3724688873146823853387

Offset: 1

Thomas Wieder, Aug 17 2008

nonn

The n labeled elements 1,2,3,...,n can be distributed in A005651(n) ways onto the levels of a single hierarchy. For the present sequence we distributed the n elements onto 1 up to n separate hierarchies. a(n) gives the number of such separate hierarchies for given n.
A hierarchy is a distribution of elements onto levels. Within a hierarchy the occupation number of level l_j is <= the occupation numbers of the levels l_i with i < j. Let the colon ":" separate two levels l_i and l_(j=i+1). Then we may have 1,2,3:4,5, but 1,2:3,4,5 is forbidden since the higher level has a greater occupation number. On the other hand, for a hierarchical ordering the second configuration is allowed.
The present sequence is the Exp transform of A005651.
The present sequence is related to A075729 which gives the number of separated hierarchical orderings. A034691 gives the number of separated hierarchical orderings for unlabeled elements. Thus we have
Hierarchies on elements:
........ unlabeled labeled
single   A000041   A005651
multiple A001970   A143463
Hierarchical orderings on elements:
........ unlabeled labeled
single   A000079   A000670
multiple A034691   A075729

			Let "|" separate two hierarchies. Then we have
n=1 gives 1 arrangement:
1
n=2 gives 4 arrangements:
1,2 1:2 2:1 1|2
n=3 gives 20 arrangements:
1,2,3 1,2:3 1:2:3 1,2|3 1:2|3 1|2|3
1,3:2 3:1:2 1,3|2 1:3|2
2,3:1 2:3:1 2,3|1 2:3|1
1:3:2 2:1|3
2:1:3 3:1|2
3:2:1 3:2|1

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..430 (first 200 terms from Alois P. Heinz)
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [From _Thomas Wieder_, Nov 14 2009]

Cf. A000041, A005651, A001970, A143463, A000079, A000670, A034691, A075729.

There is a similar formula for A075729. - Thomas Wieder, Sep 09 2008

A143463:=proc(n::integer)
# Begonnen am: 14.08.2008
# Letzte Aenderung: 14.08.2008
# Subroutines required: ListeMengenzerlegungenAuf, A005651.
local iverbose, Liste, Zerlegungen, Zerlegung, Produkt, Summe, Untermenge, ZahlElemente;
iverbose:=0;
Liste:=[seq( i, i=1..n )];
Zerlegungen:=ListeMengenzerlegungenAuf(Liste);
Summe:=0;
if iverbose=1 then
print("Zerlegungen: ",Zerlegungen);
end if;
for Zerlegung in Zerlegungen do
Produkt:=1;
if iverbose=1 then
print("Zerlegung: ",Zerlegung);
end if;
# Eine Zerlegung besteht aus Untermengen.
for Untermenge in Zerlegung do
ZahlElemente:=nops(Untermenge);
Produkt:=Produkt*A005651(ZahlElemente);
if iverbose=1 then
print("Untermenge: ",Untermenge,"A005651(ZahlElemente)",A005651(ZahlElemente));
end if;
# Ende der Schleife in der Zerlegung.
od;
Summe:=Summe+Produkt;
# Ende der Schleife ueber die Zerlegungen.
od;
print("Resultat:",Summe);
end proc;
A143463 := proc(n::integer) local k,A005651,Resultat; A005651:=array(1..20): A005651[1]:=1: A005651[2]:=3: A005651[3]:=10: A005651[4]:=47: A005651[5]:=246: A005651[6]:=1602: A005651[7]:=11481: A005651[8]:=95503: A005651[9]:=871030: A005651[10]:=8879558: A005651[11]:=98329551: A005651[12]:=1191578522: A005651[13]:=15543026747: A005651[14]:=218668538441: A005651[15]:=3285749117475: A005651[16]:=52700813279423: A005651[17]:=896697825211142: A005651[18]:=16160442591627990: A005651[19]:=307183340680888755: A005651[20]:=6147451460222703502: if n = 0 then Resultat:=1: RETURN(Resultat); end if; Resultat:=0: for k from 1 to n do Resultat:=Resultat+(A143463(n-k)*A005651[k])/((n-k)!*(k-1)!): od; Resultat:=Resultat*(n-1)!; RETURN(Resultat); end proc; [From Thomas Wieder, Sep 09 2008]
# second Maple program:
with(numtheory):
b:= proc(k) option remember; add(d/d!^(k/d), d=divisors(k)) end:
c:= proc(n) option remember; `if`(n=0, 1,
      add((n-1)!/ (n-k)!* b(k) * c(n-k), k=1..n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
       c(n) +add(binomial(n-1, k-1) *c(k) *a(n-k), k=1..n-1))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Oct 10 2008

nmax=20; Rest[CoefficientList[Series[Exp[Product[1/(1-x^k/k!),{k,1,nmax}]-1],{x,0,nmax}],x] * Range[0,nmax]!] (* Vaclav Kotesovec, May 11 2015 *)

Consider the set partitions of the n-set {1,2,...,n}. As usual, Bell(n) denotes the Bell numbers. The i-th set partition SP(i)={U(1),...,U(Z(i))} consists of Z(i) subsets U(j) with j=1,2,...,Z(i). |U(j)| is the number of elements in U(j). Then a(n) = Sum_{i=1..Bell(n)} Product_{j=1..Z(i)} A005651(|U(j)|). E.g.f.: series((1/exp(1))*exp(mul(1/(1-x^k/k!), k=1..12)), x=0,12);
a(n) = (n-1)! * Sum_{k=1..n} (a(n-k) A005651(k))/((n-k)! (k-1)!). - Thomas Wieder, Sep 09 2008
a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A005651(k)*a(n-k) and a(0)=1. - Thomas Wieder, Sep 12 2008

Partially edited by N. J. A. Sloane, Aug 24 2008

More terms from Alois P. Heinz, Oct 10 2008

A152534 Triangle T(n,k) read by rows with q-e.g.f.: 1/Product_{k>0} (1-x^k/faq(k,q)).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 3, 1, 5, 7, 11, 11, 8, 4, 1, 7, 13, 25, 36, 44, 42, 36, 24, 13, 5, 1, 11, 24, 54, 93, 142, 184, 215, 222, 208, 172, 126, 81, 44, 19, 6, 1, 15, 39, 98, 195, 344, 532, 753, 964, 1150, 1264, 1294, 1226, 1082, 880, 661, 451, 278, 151, 70, 26, 7, 1

Offset: 0

Vladeta Jovovic, Dec 06 2008

			Triangle begins:
  1;
  1;
  2,  1;
  3,  3,  3,  1;
  5,  7, 11, 11,  8,  4,  1;
  7, 13, 25, 36, 44, 42, 36, 24, 13,  5,  1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Eric Weisstein's World of Mathematics, q-Exponential Function.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A005651 (row sums), A000041 (first column), A076276 (second column), A152474, A152536.

T(n,n) gives A346980.

multinomial2q := proc(n::integer,k::integer,nparts::integer)
        local lpar ,res, constrp;
        res := [] ;
        if n< 0 or nparts <= 0 then
                ;
        elif nparts = 1 then
                if n = k then
                        return [[n]] ;
                end if;
        else
                for lpar from 0 do
                        if lpar*nparts > n or lpar > k then
                                break;
                        end if;
                        for constrp in procname(n-nparts*lpar,k-lpar,nparts-1) do
                                if nops(constrp) > 0 then
                                        res := [op(res),[op(constrp),lpar]] ;
                                end if;
                        end do:
                end do:
        end if ;
        return res ;
end proc:
multinomial2 := proc(n::integer,k::integer)
        local res,constrp ;
        res := [] ;
        for constrp in multinomial2q(n,k,n) do
                if nops(constrp) > 0 then
                        res := [op(res),constrp] ;
                end if ;
        end do:
        res ;
end proc:
faq := proc(i,q)
        mul((q^j-1)/(q-1),j=1..i) ;
end proc;
A152534 := proc(n,k)
        pi := [] ;
        for sp from 0 to n do
                pi := [op(pi),op(multinomial2(n,sp))] ;
        end do;
        tqk := 0 ;
        for p in pi do
                faqe :=1 ;
                for i from 1 to nops(p) do
                        faqe := faqe* faq(i,q)^op(i,p) ;
                end do:
                tqk := tqk+faq(n,q)/faqe ;
        end do;
        tqk ;
        coeftayl(tqk,q=0,k) ;
end proc:
for n from 1 to 8 do
        for k from 0 to binomial(n,2) do
                printf("%d,",A152534(n,k)) ;
        end do;
        printf("\n") ;
end do: # R. J. Mathar, Sep 27 2011
# second Maple program:
f:= proc(n) option remember; `if`(n<2, 1, f(n-1)*(q^n-1)/(q-1)) end:
b:= proc(n, i) option remember; simplify(`if`(n=0 or i=1, 1,
      add(b(n-i*j, i-1)/f(i)^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, q, i), i=0..degree(p)))(simplify(f(n)*b(n$2))):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 09 2021

f[n_] := f[n] = If[n < 2, 1, f[n - 1]*(q^n - 1)/(q - 1)];
b[n_, i_] := b[n, i] = Simplify[If[n == 0 || i == 1, 1,
     Sum[b[n - i*j, i - 1]/f[i]^j, {j, 0, n/i}]]];
T[n_] := CoefficientList[Simplify[f[n]*b[n, n]], q];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} faq(n,q)/Product_{i=1..n} faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1) + 2*e(2) + ... + n*e(n) = n and faq(i,q) = Product_{j=1..i} (q^j-1)/(q-1), i = 1..n.
Sum_{k=0..binomial(n,2)} T(n,k)*exp(2*Pi*I*k/n) = 1.
Sum_{k=0..binomial(n,2)} (-1)^k*T(n,k) = A152536(n). - Alois P. Heinz, Aug 09 2021

T(0,0)=1 prepended by Alois P. Heinz, Aug 09 2021

A226880 Number of n-length words w over a 10-ary alphabet {a1,a2,...,a10} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a10) >= 0, where #(w,x) counts the letters x in word w.

Original entry on oeis.org

1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, 58412751, 473076122, 3607903547, 29782240841, 241773783075, 2137404383423, 18482746670342, 173563010955990, 1554987178737075, 15169020662626702, 126731980207937625, 1160565179374262951

Offset: 0

Alois P. Heinz, Jun 21 2013

nonn

Differs from A005651 first at n=11: a(11) = 58412751 != A005651(11) = 98329551.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=10 of A226873.

b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 10):
seq(a(n), n=0..30);

A321751 Sum of coefficients of monomial symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 10, 3, 2, 1, 7, 1, 2, 2, 47, 1, 6, 1, 6, 2, 2, 1, 26, 3, 2, 10, 6, 1, 6, 1, 246, 2, 2, 2, 26, 1, 2, 2, 24, 1, 5, 1, 6, 6, 2, 1, 138, 3, 6, 2, 6, 1, 23, 2, 23, 2, 2, 1, 20, 1, 2, 7, 1602, 2, 5, 1, 6, 2, 6, 1, 105, 1, 2, 6, 6, 2, 5, 1, 114

Offset: 1

Gus Wiseman, Nov 20 2018

nonn

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

Also the number of ordered set partitions of {1, 2, ..., A001222(n)} whose blocks, when i is replaced by the i-th prime index of n, have weakly decreasing sums.

			The sum of coefficients of p(211) = m(4) + 2m(22) + 2m(31) + 2m(211) is a(12) = 7.

Wikipedia, Symmetric polynomial

Row sums of A321750.

Cf. A005651, A008277, A008480, A056239, A124794, A124795, A296150, A319182, A319225, A319226, A321742-A321765.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Times@@Factorial/@Length/@Split[Sort[Total/@s]],{s,sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[i]],{i,PrimeOmega[n]}]}],{n,50}]

A335643 Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k!).

Original entry on oeis.org

1, 1, 3, 12, 71, 462, 3890, 35133, 381583, 4411870, 58623990, 826335675, 12990713482, 216027857567, 3925135187017, 75217607162053, 1552186877466271, 33678081631793270, 778592124168964502, 18867293553102673343, 483291402186818709310, 12937553749692179771301, 363847628395565829224327

Offset: 0

Seiichi Manyama, Oct 03 2020

nonn

Cf. A005651, A335627, A335636, A335642, A336046.

max = 22; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Tan[x]^k/k!), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-tan(x)^k/k!)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, tan(x)^(i*j)/(i*j!^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} tan(x)^(i*j)/(i*(j!)^i) ).

a(n) ~ A247551 * 2^(2*n+1) * n! / Pi^(n+1). - Vaclav Kotesovec, Oct 04 2020

cp's OEIS Frontend

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A226880 Number of n-length words w over a 10-ary alphabet {a1,a2,...,a10} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a10) >= 0, where #(w,x) counts the letters x in word w.

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