A049029 -id:A049029 - cp's OEIS frontend

A092082 Triangle of numbers related to triangle A092083; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...

1, 7, 1, 91, 21, 1, 1729, 511, 42, 1, 43225, 15015, 1645, 70, 1, 1339975, 523705, 69300, 4025, 105, 1, 49579075, 21240765, 3226405, 230300, 8330, 147, 1, 2131900225, 984172735, 166428990, 13820205, 621810, 15386, 196, 1, 104463111025

Offset: 1

Wolfdieter Lang, Mar 19 2004

a(n,m) := S2(7; n,m) is the seventh triangle of numbers in the sequence S2(k;n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, respectively. a(n,1)=A008542(n), n>=1.
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 7-ary trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007
Also the Bell transform of A008542(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			{1}; {7,1}; {91,21,1}; {1729,511,42,1}; ...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, (1992), pp. 24-48.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-phys/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
W. Lang, First 10 rows.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From N. J. A. Sloane, Aug 21 2012

Cf. A092084 (row sums), A092085 (alternating row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(6*k+1, k=0..n), 9); # Peter Luschny, Jan 26 2016

mmax = 9; a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 6*x)^(-1/6))^m)/m!, {x, 0, mmax}], x^n];
Flatten[Table[a[n, m], {n, 1, mmax}, {m, 1, n}]][[1 ;; 37]] (* Jean-François Alcover, Jun 22 2011, after e.g.f. *)
rows = 9;
t = Table[Product[6k+1, {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

a(n, m) = sum(|A051151(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. with Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

a(n, m) = n!*A092083(n, m)/(m!*6^(n-m)); a(n+1, m) = (6*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

E.g.f. for m-th column: ((-1+(1-6*x)^(-1/6))^m)/m!.

A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.

Original entry on oeis.org

1, 8, 1, 120, 24, 1, 2640, 672, 48, 1, 76560, 22800, 2160, 80, 1, 2756160, 920160, 104880, 5280, 120, 1, 118514880, 43243200, 5639760, 347760, 10920, 168, 1, 5925744000, 2323918080, 336510720, 24071040, 937440, 20160, 224, 1

Offset: 1

Wolfdieter Lang Sep 14 2007

Previous name was: Triangle of numbers related to triangle A132057; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 8-ary trees. See the F. Bergeron et al. reference, especially Table 1, first row, for the e.g.f. for m=1.
a(n,m) := S2(8; n,m) is the eighth triangle of numbers in the sequence S2(k;n,m), k=1..7: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, A092082, respectively. a(n,1)=A045754(n), n>=1.

			{1}; {8,1}; {120,24,1}; {2640,672,48,1}; ...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
W. Lang, First 10 rows.

Cf. A132060 (row sums), A132061 (alternating row sums).

Cf. A092082 S2(7) triangle.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(7*k+1, k=0..n), 8); # Peter Luschny, Jan 27 2016

a[n_, m_] := a[n, m] = ((m*a[n-1, m-1]*(m-1)! + (m+7*n-7)*a[n-1, m]*m!)*n!)/(n*m!*(n-1)!);
a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1;
Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]]
(* Jean-François Alcover, Jun 17 2011 *)
rows = 8;
a[n_, m_] := BellY[n, m, Table[Product[7k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n, m) = n!*A132057(n, m)/(m!*7^(n-m)); a(n+1, m) = (7*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

E.g.f. of m-th column: ((-1+(1-7*x)^(-1/7))^m)/m!.

a(n, m) = sum(|A051186(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m):= (j, m) (Stirling2 triangle). Priv. comm. with W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

New name from Peter Luschny, Jan 27 2016

A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)(x^(5/6)d/dx)^n when n=0,2,4,6,...

Original entry on oeis.org

1, 1, 6, 7, 84, 36, 91, 1638, 1404, 216, 1729, 41496, 53352, 16416, 1296, 43225, 1296750, 2223000, 1026000, 162000, 7776, 1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656, 49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512

Offset: 1

Udita Katugampola, Mar 23 2013

			Triangle begins:
1;
1, 6;
7, 84, 36;
91, 1638, 1404, 216;
1729, 41496, 53352, 16416, 1296;
43225, 1296750, 2223000, 1026000, 162000, 7776;
1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;
49579075, 2082321150, 5354540100, 4118877000, 1300698000, 187300512, 12083904, 279936;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Even rows of A223172.

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

a[0]:= f(x):
for i from 1 to 20 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do:
for j from 1 to 10 do
b[j]:=a[2j];
end do;

A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 9, 1, 153, 27, 1, 3825, 855, 54, 1, 126225, 32895, 2745, 90, 1, 5175225, 1507815, 150930, 6705, 135, 1, 253586025, 80565975, 9205245, 499590, 13860, 189, 1, 14454403425, 4926412575, 623675430, 39180645, 1345050, 25578, 252, 1

Offset: 1

Udita Katugampola, Mar 23 2013

Also the Bell transform of A045755(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			1;
9,1;
153,27,1;
3825,855,54,1;
126225,32895,2745,90,1;
5175225,1507815,150930,6705,135,1;
253586025,80565975,9205245,499590,13860,189,1;
14454403425,4926412575,623675430,39180645,1345050,25578,252,1;

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223512-A223522, A223168-A223172, A223523-A223532.

b[0]:=g(x):
for j from 1 to 10 do
b[j]:=simplify(x^9*diff(b[j-1],x$1);
end do;
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(8*k+1, k=0..n), 10); # Peter Luschny, Jan 29 2016

rows = 8;
t = Table[Product[8k+1, {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

A223522 Triangle T(n,k) represents the coefficients of (x^20*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 20, 1, 780, 60, 1, 45240, 4320, 120, 1, 3483480, 382200, 13800, 200, 1, 334414080, 40556880, 1734600, 33600, 300, 1, 38457619200, 5039012160, 243505080, 5699400, 69300, 420, 1

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
20,1;
780,60,1;
45240,4320,120,1;
3483480,382200,13800,200,1;
334414080,40556880,1734600,33600,300,1;
38457619200,5039012160,243505080,5699400,69300,420,1;
5153320972800,718724260800,38155703040,1024322880,15262800,127680,560,1;

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522, A223168-A223172, A223523-A223532.

b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^20*diff(b[j-1],x$1);
end do;

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

A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 45, 1, 105, 180, 585, 1, 425, 2700, 2925, 9945, 1, 3075, 34650, 52650, 59670, 208845, 1, 15855, 308700, 1248975, 1253070, 1461915, 5221125, 1, 123515, 4475520, 23689575, 33972120, 35085960, 41769000

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134273.
Same partition product with length statistic is A049029.
Diagonal a(A000217) = A007696.
Row sum is A049120.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404, A157405

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-1}(-4*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

A134275 Triangle of numbers obtained from the partition array A134274.

Original entry on oeis.org

1, 5, 1, 45, 5, 1, 585, 70, 5, 1, 9945, 810, 70, 5, 1, 208845, 14895, 935, 70, 5, 1, 5221125, 284895, 16020, 935, 70, 5, 1, 151412625, 7055100, 309645, 16645, 935, 70, 5, 1, 4996616625, 192734100, 7526475, 315270, 16645, 935, 70, 5, 1, 184874815125

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(5)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			Triangle begins:
  [1];
  [5,1];
  [45,5,1];
  [585,70,5,1];
  [9945,810,70,5,1];
  ...

Wolfdieter Lang, First 10 rows and more.

Cf. A134276 (row sums). A134277 (alternating row sums).

Cf. A134151 (S2(4)').

a(n,m) = sum(product(S2(5;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S2(5;j,1)= A007696(j) = A049029(j,1) = (4*j-3)(!^4), (quadruple- or 4-factorials).

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A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)*(x^(1/2)*d/dx)^n when n is odd, and of 2^(n/2)*(x^(1/2)*d/dx)^n when n is even.

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A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)*(x^(1/6)*d/dx)^n when n is odd, and of 6^(n/2)*(x^(5/6)*d/dx)^n when n is even.

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A092082 Triangle of numbers related to triangle A092083; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...

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A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.

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A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)*(x^(5/6)*d/dx)^n when n=0,2,4,6,...

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A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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A223522 Triangle T(n,k) represents the coefficients of (x^20*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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A049353 A triangle of numbers related to triangle A030526.

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A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)(x^(1/2)d/dx)^n when n is odd, and of 2^(n/2)(x^(1/2)d/dx)^n when n is even.

A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)(x^(1/6)d/dx)^n when n is odd, and of 6^(n/2)(x^(5/6)d/dx)^n when n is even.

A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)(x^(5/6)d/dx)^n when n=0,2,4,6,...