A240172 -id:A240172 - cp's OEIS frontend

A276890 Number A(n,k) of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 6, 0, 1, 1, 3, 10, 24, 0, 1, 1, 3, 13, 44, 120, 0, 1, 1, 3, 13, 62, 234, 720, 0, 1, 1, 3, 13, 75, 352, 1470, 5040, 0, 1, 1, 3, 13, 75, 466, 2348, 10656, 40320, 0, 1, 1, 3, 13, 75, 541, 3272, 17880, 87624, 362880, 0

Offset: 0

Alois P. Heinz, Sep 21 2016

Column k > 0 is asymptotic to exp(k-1) * n!. - Vaclav Kotesovec, Sep 22 2016

			Square array A(n,k) begins:
  1,    1,     1,     1,     1,     1,     1,     1, ...
  0,    1,     1,     1,     1,     1,     1,     1, ...
  0,    2,     3,     3,     3,     3,     3,     3, ...
  0,    6,    10,    13,    13,    13,    13,    13, ...
  0,   24,    44,    62,    75,    75,    75,    75, ...
  0,  120,   234,   352,   466,   541,   541,   541, ...
  0,  720,  1470,  2348,  3272,  4142,  4683,  4683, ...
  0, 5040, 10656, 17880, 26032, 34792, 42610, 47293, ...

Alois P. Heinz, Antidiagonals n = 0..42, flattened

Columns k=0-10: A000007, A000142, A240172, A276893, A276894, A276895, A276896, A276897, A276898, A276899, A276900.
Main diagonal gives: A000670.
Cf. A276719, A276891.

b:= proc(n, m, l) option remember; `if`(n=0, m!,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0),
             `if`(k=1, n!, b(n, 0, [0$(k-1)]))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, m!, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[k==0, If[n==0, 1, 0], If[k==1, n!, b[n, 0, Array[0&, k-1]]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..k} A276891(n,j).

A240921 G.f.: Sum_{n>=0} n^n * x^n * (1 + nx)^n / (1 + nx + n^2*x^2)^(n+1).

Original entry on oeis.org

1, 1, 3, 18, 146, 1530, 19620, 297360, 5201784, 103146120, 2286181800, 56011087440, 1503057473280, 43844234353920, 1381310964633600, 46743301840435200, 1690919874777893760, 65116170597269151360, 2659604669692822051200, 114838104572526535200000, 5226654647185285702752000

Offset: 0

Paul D. Hanna, Aug 02 2014

nonn

a(n) is divisible by [n/2]!.
Compare definition to the following identity, which holds for all fixed k:
Sum_{n>=0} n!*x^n = Sum_{n>=0} x^n * (n + k*x)^n / (1 + n*x + k*x^2)^(n+1).

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 146*x^4 + 1530*x^5 + 19620*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2)^2 + 2^2*x^2*(1+2*x)^2/(1+2*x+4*x^2)^3 + 3^3*x^3*(1+3*x)^3/(1+3*x+9*x^2)^4 + 4^4*x^4*(1+4*x)^4/(1+4*x+16*x^2)^5 +...

Cf. A240172, A240956, A240957, A240958, A240921, A008277 (Stirling2).

Table[Sum[(n-k)! * StirlingS2[n, n-k] * Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2014 *)

/* From definition: */
{a(n)=local(A=1); A=sum(m=0, n, m^m*x^m*(1+m*x)^m/(1 + m*x + m^2*x^2 +x*O(x^n))^(m+1)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

/* From formula for a(n): */
{Stirling2(n, k)=sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!}
{a(n)=sum(k=0,n\2, (n-k)!*Stirling2(n, n-k)*binomial(n-k,k))}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..[n/2]} (n-k)! * Stirling2(n, n-k) * binomial(n-k,k).

a(n) ~ c * d^n * n^n / exp(n), where d = r*(-1+3*r-3*r^2)/(1-3*r+2*r^2) = 2.334305682349197638435662..., where r = 0.722795640379451585372396... is the root of the equation (1-r) * (r + 1/LambertW(-exp(-1/r)/r)) + (2*r-1)^2 = 0, and c = 1.04764685950245700560418116727397... . - Vaclav Kotesovec, Aug 03 2014

A370510 Expansion of Sum_{k>=0} k! * ( x * (1+x^3) )^k.

Original entry on oeis.org

1, 1, 2, 6, 25, 124, 738, 5136, 40922, 367218, 3664224, 40240560, 482278326, 6263414736, 87618506160, 1313435465280, 21003904630824, 356910121855320, 6422020465846320, 121980351190294800, 2438956634267865720, 51206322309647263200, 1126314497201852150640

Offset: 0

Seiichi Manyama, Feb 20 2024

Cf. A240172, A370509.

Cf. A370508.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1+x^3))^k))

a(n) = sum(k=0, n\4, (n-3*k)!*binomial(n-3*k, k));

a(n) = Sum_{k=0..floor(n/4)} (n-3*k)! * binomial(n-3*k,k).

A370509 Expansion of Sum_{k>=0} k! * ( x * (1+x^2) )^k.

Original entry on oeis.org

1, 1, 2, 7, 28, 138, 818, 5658, 44784, 399366, 3962256, 43289760, 516432984, 6679346280, 93091875120, 1390851720840, 22175338353120, 375794883339120, 6745177713093840, 127830886641354960, 2550687440585679360, 53451172032327664560, 1173650135526055272960

Offset: 0

Seiichi Manyama, Feb 20 2024

Cf. A240172, A370510.

Cf. A212580.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1+x^2))^k))

a(n) = sum(k=0, n\3, (n-2*k)!*binomial(n-2*k, k));

a(n) = Sum_{k=0..floor(n/3)} (n-2*k)! * binomial(n-2*k,k).

A346550 Expansion of Sum_{k>=0} k! * x^k * (1 + x)^(k+1).

Original entry on oeis.org

1, 2, 4, 13, 54, 278, 1704, 12126, 98280, 893904, 9017280, 99918120, 1206500400, 15768729360, 221792780160, 3340515069360, 53641756586880, 914849722725120, 16514863528665600, 314599179867396480, 6306817346711481600, 132727279189258656000

Offset: 0

Seiichi Manyama, Nov 30 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A184185, A240172.

a[n_] := Sum[k! * Binomial[k + 1, n - k], {k, Floor[n/2], n}]; Array[a, 22, 0] (* Amiram Eldar, Nov 30 2021 *)

a(n) = sum(k=n\2, n, k!*binomial(k+1, n-k));

a(n) = if(n<3, 2^n, (n-2)*a(n-1)+2*(n-1)*a(n-2)+(n-2)*a(n-3));

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k*(1+x)^(k+1)))

a(n) = Sum_{k=floor(n/2)..n} k! * binomial(k+1,n-k).
a(n) = A240172(n-1) + A240172(n) for n > 0.
a(n) = (n-2) * a(n-1) + 2 * (n-1) * a(n-2) + (n-2) * a(n-3) for n > 2.
a(n) ~ exp(1) * n! * (1 - 1/n + 3/(2*n^2) - 2/(3*n^3) - 47/(24*n^4) + 49/(120*n^5) + 6421/(720*n^6) + ...). - Vaclav Kotesovec, Dec 11 2021

A370668 Expansion of Sum_{k>0} k! * ( x * (1+x^k) )^k.

Original entry on oeis.org

1, 3, 6, 28, 120, 740, 5040, 40416, 362898, 3629400, 39916800, 479006070, 6227020800, 87178326480, 1307674369200, 20922790210656, 355687428096000, 6402373709004720, 121645100408832000, 2432902008212929224, 51090942171709545840, 1124000727778046764800

Offset: 1

Seiichi Manyama, Feb 25 2024

nonn

Cf. A240172, A370509, A370510.

Cf. A217668.

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*(x*(1+x^k))^k))

a(n) = sumdiv(n,d, d!*binomial(d, n/d-1));

a(n) = Sum_{d|n} d! * binomial(d,n/d-1).

If p is an odd prime, a(p) = p!.

cp's OEIS Frontend

A276890 Number A(n,k) of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A240921 G.f.: Sum_{n>=0} n^n * x^n * (1 + n*x)^n / (1 + n*x + n^2*x^2)^(n+1).

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A370510 Expansion of Sum_{k>=0} k! * ( x * (1+x^3) )^k.

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A370509 Expansion of Sum_{k>=0} k! * ( x * (1+x^2) )^k.

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A346550 Expansion of Sum_{k>=0} k! * x^k * (1 + x)^(k+1).

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A370668 Expansion of Sum_{k>0} k! * ( x * (1+x^k) )^k.

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A240921 G.f.: Sum_{n>=0} n^n * x^n * (1 + nx)^n / (1 + nx + n^2*x^2)^(n+1).