A102221 -id:A102221 - cp's OEIS frontend

A192723 Erroneous version of A102221.

1, 5, 55, 1077, 32951, 1451723, 87054773, 6818444405, 675900963271, 82717196780955, 12248810651651332, 2158585005685222202, 446445657799551722026, 107087164031952012004954

Offset: 1

dead

A336195 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^3 * a(k).

Original entry on oeis.org

1, 1, 9, 271, 19353, 2699251, 650553183, 248978967973, 142238892608025, 115699539306013867, 129097362200437841259, 191726066802105786953113, 369666963359241578736653775, 906204961889202975320635813201, 2774573804997927027583123365125685

Offset: 0

Ilya Gutkovskiy, Jul 11 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..180

Column k=3 of A326322.

Cf. A000670, A102221, A336196, A336197.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^3 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^3, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^3

a(n) = (n!)^3 * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^3).

A326322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = sum of the k-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 13, 8, 1, 1, 9, 55, 75, 16, 1, 1, 17, 271, 1077, 541, 32, 1, 1, 33, 1459, 19353, 32951, 4683, 64, 1, 1, 65, 8263, 395793, 2699251, 1451723, 47293, 128, 1, 1, 129, 48115, 8718945, 262131251, 650553183, 87054773, 545835, 256

Offset: 0

Alois P. Heinz, Sep 11 2019

For k>=1, A(n,k) is the number of k-tuples (p_1,p_2,...,p_k) of ordered set partitions of [n] such that the sequence of block lengths in each ordered partition p_i is identical. Equivalently, A(n,k) is the number of chains from s to t where [s,t] is any n-interval in the binomial poset B_k = B*B*...*B (k times), B is the lattice of all finite subsets of {1,2,...} ordered by inclusion and * is the Segre product. See Stanley reference. - Geoffrey Critzer, Dec 16 2020

			A(2,2) = M(2; 2)^2 + M(2; 1,1)^2 = 1 + 4 = 5.
Square array A(n,k) begins:
   1,   1,     1,       1,         1,           1, ...
   1,   1,     1,       1,         1,           1, ...
   2,   3,     5,       9,        17,          33, ...
   4,  13,    55,     271,      1459,        8263, ...
   8,  75,  1077,   19353,    395793,     8718945, ...
  16, 541, 32951, 2699251, 262131251, 28076306251, ...

R. P. Stanley, Enumerative Combinatorics, Vol. I, second edition, Example 3.18.3d page 322.

Alois P. Heinz, Antidiagonals n = 0..60, flattened
Wikipedia, Multinomial coefficients

Columns k=0-2 give: A011782, A000670, A102221.
Rows n=0+1, 2 give A000012, A000051.
Main diagonal gives A326321.
Cf. A183610.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-i, k)/i!^k, i=1..n))
    end:
A:= (n, k)-> n!^k*b(n, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n, j)^k*A(j, k), j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n, j]^k A[j, k], {j, 0, n-1}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten  (* Jean-François Alcover, Dec 03 2020, after 2nd Maple program *)

Let E_k(x) = Sum_{n>=0} x^n/n!^k. Then 1/(2-E_k(x)) = Sum_{n>=0} A(n,k)*x^n/n!^k. - Geoffrey Critzer, Dec 16 2020

A102220 Triangular matrix, read by rows, equal to [2*I - A008459]^(-1), i.e., the matrix inverse of the difference of twice the identity matrix and the triangular matrix of squared binomial coefficients.

Original entry on oeis.org

1, 1, 1, 5, 4, 1, 55, 45, 9, 1, 1077, 880, 180, 16, 1, 32951, 26925, 5500, 500, 25, 1, 1451723, 1186236, 242325, 22000, 1125, 36, 1, 87054773, 71134427, 14531391, 1319325, 67375, 2205, 49, 1, 6818444405, 5571505472, 1138150832, 103334336, 5277300, 172480, 3920, 64, 1

Offset: 0

Paul D. Hanna, Dec 31 2004

Column 0 forms A102221. Row sums form twice column 0 for n>0. Matrix logarithm is A102222.

			Rows begin:
[1],
[1,1],
[5,4,1],
[55,45,9,1],
[1077,880,180,16,1],
[32951,26925,5500,500,25,1],
[1451723,1186236,242325,22000,1125,36,1],...
and equal the term-by-term product of column 0
with the squared binomial coefficients (A008459):
[(1)1^2],
[(1)1^2,(1)1^2],
[(5)1^2,(1)2^2,(1)1^2],
[(55)1^2,(5)3^2,(1)3^2,(1)1^2],
[(1077)1^2,(55)4^2,(5)6^2,(1)4^2,(1)1^2],...
The matrix inverse is [2*I - A008459]:
[1],
[ -1,1],
[ -1,-4,1],
[ -1,-9,-9,1],
[ -1,-16,-36,-16,1],...

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

Cf. A008459, A102221, A102222.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n, i)/i!, i=1..n))
    end:
T:= (n, k)-> binomial(n, k)^2*b(n-k)*(n-k)!:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Sep 10 2019

nmax = 10;
M = Inverse[2 IdentityMatrix[nmax+1] - Table[Binomial[n, k]^2, {n, 0, nmax}, {k, 0, nmax}]];
T[n_, k_] := M[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)

{T(n,k)=(matrix(n+1,n+1,i,j,if(i==j,2,0)-binomial(i-1,j-1)^2)^-1)[n+1,k+1]}

T(n,k) = C(n,k)^2*A102221(n-k). T(n,0) = A102221(n). 2*A102221(n) = Sum_{k=0..n} T(n,k) for n>0.

A336196 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^4 * a(k).

Original entry on oeis.org

1, 1, 17, 1459, 395793, 262131251, 359993423843, 915919888063853, 3975467425523532305, 27639424688447366285203, 292886774320942590679779267, 4544030770812055230064359134573, 99847457331663057820508375752459491, 3021907600842518917755426740899056448141

Offset: 0

Ilya Gutkovskiy, Jul 11 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..143

Column k=4 of A326322.

Cf. A000670, A102221, A336195, A336197.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^4 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^4, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^4

a(n) = (n!)^4 * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^4).

A336197 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^5 * a(k).

Original entry on oeis.org

1, 1, 33, 8263, 8718945, 28076306251, 224968772934303, 3896175006605313013, 131557135159637950535265, 8004845815916146011992853811, 824857614282973828473497207276283, 136888961901974254918775560412316183913, 35099479542762449254288789631427310686677535

Offset: 0

Ilya Gutkovskiy, Jul 11 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..120

Column k=5 of A326322.

Cf. A000670, A102221, A336195, A336196.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^5 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
nmax = 12; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^5, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^5

a(n) = (n!)^5 * [x^n] 1 / (1 - Sum_{k>=1} x^k / (k!)^5).

A336258 a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.

Original entry on oeis.org

1, 1, 5, 58, 1208, 39476, 1861372, 119587224, 10040970816, 1067383279872, 140110136642304, 22256626639796352, 4207858001708629248, 933704296260740939520, 240293228328619963492608, 70992050129486593239246336, 23863916105454465092261412864

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..247

Cf. A007840, A074707, A102221, A336243, A336259, A336260, A336261.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)/i^2, i=1..n))
    end:
a:= n-> n!^2*b(n):
seq(a(n), n=0..16);  # Alois P. Heinz, Jan 04 2024

a[0] = 1; a[n_] := a[n] = (n!)^2 Sum[a[k]/(k! (n - k))^2, {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - PolyLog[2, x]), {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * [x^n] 1 / (1 - polylog(2,x)).

a(n) ~ (n!)^2 / (-log(1-r) * r^n), where r = 0.76154294453204558806805187241... is the root of the equation polylog(2,r) = 1. - Vaclav Kotesovec, Jul 15 2020

A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

Original entry on oeis.org

1, 1, 4, 1, 18, 36, 1, 68, 432, 576, 1, 250, 3900, 14400, 14400, 1, 922, 32400, 252000, 648000, 518400, 1, 3430, 262542, 3880800, 19404000, 38102400, 25401600, 1, 12868, 2119152, 56664384, 493920000, 1795046400, 2844979200, 1625702400

Offset: 1

Peter Bala, Jul 11 2011

Compare with triangle A019538, whose entries are given by
... Sum multinomial(n; n_1,n_2,...,n_k), where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.
For related tables see A061691 and A192721.
Let P be the poset of all ordered pairs (S,T) of subsets of [n] with |S|=|T|, ordered componentwise by inclusion.  T(n,k) is the number of length k chains in P from ({},{}) to ([n],[n]). - Geoffrey Critzer, Apr 15 2020

			The triangle begins
n/k|..1.....2.......3........4........5........6
================================================
.1.|..1
.2.|..1.....4
.3.|..1....18.....36
.4.|..1....68.....432......576
.5.|..1...250....3900....14400....14400
.6.|..1...922...32400...252000...648000...518400
...
T(4,2) = 68:
There are 3 compositions of 4 into 2 parts, namely, 4 = 2 + 2 = 1 + 3 = 3 + 1; hence
T(4,2) = (4!/(2!*2!))^2 + (4!/(1!*3!))^2 + (4!/(3!*1!))^2
= 36 + 16 + 16 = 68.
Matrix identity: A192721 * Pascal's triangle = row reverse of A192722:
/...1................\ /..1..............\
|...3.....1...........||..1....1..........|
|..19....16.....5.....||..1....2....1.....|
|.211...299....65....1||..1....3....3....1|
|.....................||..................|
=
/...1...................\
|...4......1.............|
|..36.....18......1......|
|.576....432.....68.....1|
|........................|

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A001044, A002190, A061691, A192721, A102221 (row sums), A000275 (alternating row sums).

J := unapply(BesselJ(0, 2*sqrt(-1)*sqrt(z)), z):
G := 1/(1-x*(J(z)-1)):
Gser := simplify(series(G, z = 0, 15)):
for n from 1 to 14 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 14 do seq(coeff(P[n], x, k), k = 1..n) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(
      `if`(n=0, 1, add(x*b(n-i)/i!^2, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)*n!^2):
seq(T(n), n=1..14);  # Alois P. Heinz, Sep 10 2019

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[x b[n-i]/i!^2, {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n] n!^2];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Generating function: Let J(z) = Sum_{n>=0} z^n/n!^2. Then
1 + Sum_{n>=1} (Sum_{k = 1..n} T(n,k)*x^k)*z^n/n!^2 = 1/(1 - x*(J(z) - 1))
  = 1 + x*z + (x + 4*x^2)*z^2/2!^2 + (x + 18*x^2 + 36*x^3)*z^3/3!^2 + ....
Relations with other sequences:
The change of variable z -> z/x followed by x -> 1/(x - 1) transforms the above bivariate generating function 1/(1 - x*(J(z) - 1)) into (1 - x)/(-x + J(z*(x-1))), which is the generating function for A192721.
1/k!*T(n,k) = A061691(n,k).
T(n,n) = n!^2 = A001044(n).
Row sums = A102221.
For n>=1, Sum_{k = 1..n} (-1)^(n+k)*T(n,k)/k = A002190(n).

A342196 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^2 * a(k-1).

Original entry on oeis.org

1, 1, 5, 23, 155, 1355, 14371, 183911, 2781283, 48726355, 976903875, 22183097191, 565060532965, 16016170519017, 501714014484813, 17265124180702953, 649178961366102597, 26544344366333824055, 1175291769917975444817, 56133021061270139242637, 2881893164859601701738005

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A023998, A040027, A101514, A102221, A342197, A342198, A342199.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 20}]

A340886 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 2^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 6, 76, 1720, 60816, 3096384, 214579296, 19422473088, 2224980891904, 314675568756736, 53849929134122496, 10966912240761425920, 2621246193301011159040, 726608751113679704248320, 231217063994112487051984896, 83713709650818121936828858368

Offset: 0

Ilya Gutkovskiy, Jan 25 2021

nonn

Cf. A102221, A122704, A340887, A340888.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 2^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[2/(3 - BesselI[0, 2 Sqrt[2 x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 2 / (3 - BesselI(0,2*sqrt(2*x))).

cp's OEIS Frontend

A192723 Erroneous version of A102221.

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A336195 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^3 * a(k).

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A326322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) = sum of the k-th powers of multinomials M(n; mu), where mu ranges over all compositions of n.

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A102220 Triangular matrix, read by rows, equal to [2*I - A008459]^(-1), i.e., the matrix inverse of the difference of twice the identity matrix and the triangular matrix of squared binomial coefficients.

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A336196 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^4 * a(k).

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A336197 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^5 * a(k).

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A336258 a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.

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A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

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