A105752 -id:A105752 - cp's OEIS frontend

A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

1, 3, 1, 12, 9, 1, 60, 75, 18, 1, 360, 660, 255, 30, 1, 2520, 6300, 3465, 645, 45, 1, 20160, 65520, 47880, 12495, 1365, 63, 1, 181440, 740880, 687960, 235305, 35700, 2562, 84, 1, 1814400, 9072000, 10372320, 4452840, 877905, 86940, 4410, 108, 1

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A030523.
a(n,1)= A001710(n+1). a(n,m)=: S1p(3; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035342(n,m) := S2(3; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+2 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
a(4,2)=75=4*(3*4)+3*(3*3) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*3*4)=12 colored versions, e.g. ((1c1),(2c1,3c3,4c2)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 3 colors, c1, c2 and c3, can be chosen and the vertex labeled 4 with j=2 can come in 4 colors, e.g. c1, c2, c3 and c4. Therefore there are 4*(1)*(1*3*4)=48 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*3)*(1*3))=27 such forests, e.g. ((1c1,3c2)(2c1,4c1)) or ((1c1,3c2)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001710(n+2) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
  [1],
  [3, 1],
  [12, 9, 1],
  [60, 75, 18, 1],
  [360, 660, 255, 30, 1],
  [2520, 6300, 3465, 645, 45, 1],
  ...

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
John Riordan, Letter, Apr 28 1976.

Cf. A001710, A049376, A105752.

Alternating row sums A134138.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (2m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
a[n_, k_] := -(-1/2)^k*(n+1)!*HypergeometricPFQ[{1-k, n/2+1, (n+3)/2}, {3/2, 2}, 1]/(k-1)!; Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 28 2013, after Vladimir Kruchinin *)
a[0] = 0; a[n_] := (n + 1)!/2;
T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, a[0]^n], Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 19 2016, after Peter Luschny, updated Jan 01 2021 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[(k+2)!/2, {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1),j,1,k))/(2^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+2)//2, 9) # Peter Luschny, Jan 19 2016

a(n, m) = n!*A030523(n, m)/(m!*2^(n-m)); a(n, m) = (2*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n, m) = sum(|S1(n, j)|* A075497(j, m), j=m..n) (matrix product), with S1(n, j) := A008275(n, j) (signed Stirling1 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference.

a(n, k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+2*j-1,2*j-1)))/(2^k*k!) - Vladimir Kruchinin, Apr 01 2011

New name from Peter Luschny, Jan 19 2016

A357119 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} |Stirling1(n,k*j)|.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 6, 0, 1, 0, 0, 3, 24, 0, 1, 0, 0, 1, 12, 120, 0, 1, 0, 0, 0, 6, 60, 720, 0, 1, 0, 0, 0, 1, 35, 360, 5040, 0, 1, 0, 0, 0, 0, 10, 226, 2520, 40320, 0, 1, 0, 0, 0, 0, 1, 85, 1645, 20160, 362880, 0, 1, 0, 0, 0, 0, 0, 15, 735, 13454, 181440, 3628800, 0

Offset: 0

Seiichi Manyama, Oct 17 2022

			Square array begins:
  1,   1,   1,   1,  1,  1, 1, ...
  0,   1,   0,   0,  0,  0, 0, ...
  0,   2,   1,   0,  0,  0, 0, ...
  0,   6,   3,   1,  0,  0, 0, ...
  0,  24,  12,   6,  1,  0, 0, ...
  0, 120,  60,  35, 10,  1, 0, ...
  0, 720, 360, 226, 85, 15, 1, ...

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Columns k=0-3 give: A000007, A000142, (-1)^n * A105752(n), A357828.

Cf. A357293.

T(n, k) = sum(j=0, n, abs(stirling(n, k*j, 1)));

T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (-log(1-x+x*O(x^n)))^(k*j)/(k*j)!), n));

Pochhammer(x, n) = prod(k=0, n-1, x+k);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Pochhammer(w^j, n)))/k);

For k > 0, e.g.f. of column k: Sum_{j>=0} (-log(1-x))^(k*j)/(k*j)!.

For k > 0, T(n,k) = ( Sum_{j=0..k-1} (w^j)_n )/k, where (x)_n is the Pochhammer symbol and w = exp(2*Pi*i/k).

A357683 a(n) = Sum_{k=0..floor(n/2)} n^k * |Stirling1(n,2*k)|.

Original entry on oeis.org

1, 0, 2, 9, 60, 500, 4920, 55566, 706720, 9979200, 154706760, 2609691700, 47547916416, 929943488448, 19421810408000, 431196538865400, 10137091700736000, 251485260368396288, 6563768030597826720, 179746132716715050000, 5152012082327932518400

Offset: 0

Seiichi Manyama, Oct 09 2022

nonn

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A105752, A263687.

a(n) = sum(k=0, n\2, n^k*abs(stirling(n, 2*k, 1)));

a(n) = round(n!*polcoef(cosh(sqrt(n)*log(1-x+x*O(x^n))), n));

a(n) = round((prod(k=0, n-1, sqrt(n)+k)+prod(k=0, n-1, -sqrt(n)+k)))/2;

a(n) = n! * [x^n] cosh( sqrt(n) * log(1-x) ).
a(n) = ( (sqrt(n))_n + (-sqrt(n))_n )/2, where (x)_n is the Pochhammer symbol.
a(n) ~ n^(n + sqrt(n)/2 - 1/4) / (2*exp(n - sqrt(n) - 1/2)) * (1 - 3/(4*sqrt(n))). - Vaclav Kotesovec, Oct 10 2022

A357712 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * log(1-x) ).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 12, 0, 1, 0, 4, 9, 26, 60, 0, 1, 0, 5, 12, 42, 140, 360, 0, 1, 0, 6, 15, 60, 240, 896, 2520, 0, 1, 0, 7, 18, 80, 360, 1614, 6636, 20160, 0, 1, 0, 8, 21, 102, 500, 2520, 12474, 55804, 181440, 0, 1, 0, 9, 24, 126, 660, 3620, 20160, 108900, 525168, 1814400, 0

Offset: 0

Seiichi Manyama, Oct 10 2022

			Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  0,   0,   0,   0,   0, ...
  0,  1,   2,   3,   4,   5, ...
  0,  3,   6,   9,  12,  15, ...
  0, 12,  26,  42,  60,  80, ...
  0, 60, 140, 240, 360, 500, ...

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.
Main diagonal gives A357683.
Cf. A357681.

T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));

T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;

T(n,k) = Sum_{j=0..floor(n/2)} k^j * |Stirling1(n,2*j)|.
T(n,k) = ( (sqrt(k))_n + (-sqrt(k))_n )/2, where (x)_n is the Pochhammer symbol.
T(0,k) = 1, T(1,k) = 0; T(n,k) = (2*n-3) * T(n-1,k) - (n^2-4*n+4-k) * T(n-2,k).

A357834 a(n) = Sum_{k=0..floor(n/3)} Stirling1(n,3*k).

Original entry on oeis.org

1, 0, 0, 1, -6, 35, -224, 1603, -12810, 113589, -1109472, 11852841, -137611110, 1726238787, -23277264192, 335861699355, -5164348236138, 84316474011861, -1456893047937600, 26562992204112273, -509679388313669574, 10266675502780006947, -216625348636705401120

Offset: 0

Seiichi Manyama, Oct 14 2022

sign

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A357835, A357836.

Cf. A105752.

a(n) = sum(k=0, n\3, stirling(n, 3*k, 1));

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

Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-1, n)+Pochhammer(-w, n)+Pochhammer(-w^2, n))/3;

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(log(1+x)).

a(n) = (-1)^n * ( (-1)_n + (-w)_n + (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.

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A046089 Triangle read by rows, the Bell transform of (n+2)!/2 without column 0.

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A357119 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} |Stirling1(n,k*j)|.

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A357683 a(n) = Sum_{k=0..floor(n/2)} n^k * |Stirling1(n,2*k)|.

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A357712 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * log(1-x) ).

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A357834 a(n) = Sum_{k=0..floor(n/3)} Stirling1(n,3*k).

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