A049378 -id:A049378 - cp's OEIS frontend

A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)binomial(-k,j-1)x^j/j).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 73, 120, 1, 1, 1, 6, 33, 154, 501, 720, 1, 1, 1, 7, 46, 273, 1306, 4051, 5040, 1, 1, 1, 8, 61, 436, 2721, 12976, 37633, 40320, 1, 1, 1, 9, 78, 649, 4956, 31701, 147484, 394353, 362880, 1

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array B(j,k) begins:
   1,   1,   1,    1,    1, ...
   0,   1,   2,    3,    4, ...
   0,   1,   3,    6,   10, ...
   0,   1,   4,   10,   20, ...
   0,   1,   5,   15,   35, ...
   0,   1,   6,   21,   56, ...
Square array A(n,k) begins:
   1,   1,   1,    1,    1, ...
   1,   1,   1,    1,    1, ...
   1,   2,   3,    4,    5, ...
   1,   6,  13,   22,   33, ...
   1,  24,  73,  154,  273, ...
   1, 120, 501, 1306, 2721, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..10 give A000012, A000142, A000262, A049376, A049377, A049378, A049402, A132164, A293986, A293987, A293988.
Rows n=0-1 give A000012.
Main diagonal gives A293989.
Cf. A293012, A293991.

B[j_, k_] := (-1)^(j-1)*Binomial[-k, j-1];
A[0, ] = 1; A[n, k_] := (n-1)!*Sum[B[j, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A049353 A triangle of numbers related to triangle A030526.

Original entry on oeis.org

1, 5, 1, 30, 15, 1, 210, 195, 30, 1, 1680, 2550, 675, 50, 1, 15120, 34830, 14025, 1725, 75, 1, 151200, 502740, 287280, 51975, 3675, 105, 1, 1663200, 7692300, 5961060, 1482705, 151200, 6930, 140, 1, 19958400, 124740000, 126913500, 41545980

Offset: 1

Wolfdieter Lang

a(n,1)= A001720(n+3). a(n,m)=: S1p(5; 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), S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352(n,m).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049029(n,m) := S2(5; 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+4 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001720. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
  {1};
  {5,1};
  {30,15,1}; E.g., row polynomial E(3,x)=30*x+15*x^2+x^3.
  {210,195,30,1};
  ...
a(4,2)= 195 =4*(5*6)+3*(5*5) 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*5*6)=30 colored versions, e.g., ((1c1),(2c1,3c5,4c6)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 5 colors, c1..c5, can be chosen and the vertex labeled 4 with j=2 can come in 6 colors, e.g., c1..c6. Therefore there are 4*((1)*(1*5*6))=120 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*5)*(1*5))=75 such forests, e.g., ((1c1,3c4)(2c1,4c5)) or ((1c1,3c5)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Cf. A049378 (row sums).

Cf. A134139 (alternating row sums).

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+4)!/24, 10); # Peter Luschny, Jan 28 2016

a[n_, m_] /; n >= m >= 1 := a[n, m] = (4m + 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 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+4)!/24&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+4*j-1,4*j-1),j,1,k))/(4^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n, m) = n!*A030526(n, m)/(m!*4^(n-m)); a(n, m) = (4*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n,k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+4*j-1,4*j-1)))/(4^k*k!). - Vladimir Kruchinin, Apr 01 2011

A157385 A partition product of Stirling_1 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 30, 1, 105, 120, 210, 1, 425, 1800, 1050, 1680, 1, 3075, 18600, 18900, 10080, 15120, 1, 15855, 174300, 338100, 211680, 105840, 151200, 1, 123515, 2227680, 4865700, 4327680, 2540160, 1209600, 1663200, 1, 757755

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144355.
Same partition product with length statistic is A049353.
Diagonal a(A000217(n)) = rising_factorial(5,n-1), A001720(n+3).
Row sum is A049378.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.

Cf. A157386, A157384, A157383, A157400, A126074, A157391, A157392, A157393, A157394, A157395

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-3).

A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 30, 15, 1, 210, 120, 75, 30, 1, 1680, 1050, 1500, 300, 375, 50, 1, 15120, 10080, 15750, 9000, 3150, 9000, 1875, 600, 1125, 75, 1, 151200, 105840, 176400, 220500, 35280, 110250, 63000, 78750, 7350, 31500, 13125, 1050, 2625, 105, 1, 1663200, 1209600, 2116800

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(5;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Fifth member (K=5) in the family M31(K) of partition number arrays.
If M31(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(5)|:= A049353.

			[1];[5,1];[30,15,1];[210,120,75,30,1];[1680,1050,1500,300,375,50,1];...
a(4,3)= 75 = 3*|S1(5;2,1)|^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A049378 (row sums).

A144354 (M31(4) array), A144356 (M31(6) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(5;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)|= A001720(n+3) = (n+3)!/4!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k)=A036040.

A134139 Alternating row sums of triangle A049353 (S1p(5)).

Original entry on oeis.org

1, 4, 16, 44, -244, -7336, -112664, -1406336, -14058944, -66627136, 1879992896, 85070733824, 2359547577856, 54106723270144, 1055100890838016, 15611883969810944, 48348082318102016, -9055743653514520576, -521351156412528902144

Offset: 1

Wolfdieter Lang Oct 12 2007

Cf. A049378 (row sums of A049353).

a(n)=sum(A049353(n,m)*(-1)^(m-1),m=1..n), n>=1.

E.g.f.: 1-exp(-x*(2-x)*(2-2*x+x^2)/(4*(1-x)^4)). Cf. e.g.f. first column of A049353.

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A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)*binomial(-k,j-1)*x^j/j).

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A049353 A triangle of numbers related to triangle A030526.

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A157385 A partition product of Stirling_1 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

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A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).

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A134139 Alternating row sums of triangle A049353 (S1p(5)).

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A291709 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)binomial(-k,j-1)x^j/j).