A008297 -id:A008297 - cp's OEIS frontend

A079639 Matrix product of Stirling1-triangle A008275(n,k) and unsigned Lah-triangle |A008297(n,k)|.

1, 1, 1, 2, 3, 1, 4, 11, 6, 1, 14, 40, 35, 10, 1, 38, 184, 195, 85, 15, 1, 216, 840, 1204, 665, 175, 21, 1, 600, 4920, 7616, 5369, 1820, 322, 28, 1, 6240, 26616, 54116, 44016, 18669, 4284, 546, 36, 1, 9552, 197856, 392460, 383480, 191205, 54453, 9030, 870, 45, 1, 319296, 1177176, 3229776, 3449600, 2017070, 679371, 139293, 17490, 1320, 55, 1, -519312

Offset: 1

Vladeta Jovovic, Jan 30 2003

Also the Bell transform of A006252(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Cf. A006252 (first column).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add(k!*combinat:-stirling1(n+1,k),k=0..n+1),9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, Sum[k!*StirlingS1[n+1, k], {k, 0, n+1}]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 27 2018, after Peter Luschny *)

T(n, k) = Sum_{i=k..n} A008275(n, i) * |A008297(i, k)|.

E.g.f: (1+x)^(y/(1-log(1+x))). - Vladeta Jovovic, Nov 22 2003

A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

Original entry on oeis.org

1, 3, 1, 13, 9, 1, 75, 79, 18, 1, 541, 765, 265, 30, 1, 4683, 8311, 3870, 665, 45, 1, 47293, 100989, 59101, 13650, 1400, 63, 1, 545835, 1362439, 960498, 278901, 38430, 2618, 84, 1, 7087261, 20246445, 16700545, 5844510, 1012431, 92610, 4494, 108, 1

Offset: 1

Vladeta Jovovic, Nov 22 2003

Also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Also the number of k-dimensional flats of the n-dimensional Catalan arrangement. - Shuhei Tsujie, May 05 2019

N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000670(first column), A075729(row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add(combinat:-eulerian1(n+1, k)*2^k, k=0..n+1), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, HurwitzLerchPhi[1/2, -n-1, 0]/2], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 27 2018, after Peter Luschny *)

E.g.f.: exp((exp(x)-1)*y/(2-exp(x))).

A088814 Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling2-triangle A008277(n,k).

Original entry on oeis.org

1, 3, 1, 13, 9, 1, 73, 79, 18, 1, 501, 755, 265, 30, 1, 4051, 7981, 3840, 665, 45, 1, 37633, 93135, 57631, 13580, 1400, 63, 1, 394353, 1192591, 911582, 274141, 38290, 2618, 84, 1, 4596553, 16645431, 15285313, 5633922, 999831, 92358, 4494, 108, 1, 58941091

Offset: 1

Vladeta Jovovic, Nov 22 2003

Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Cf. A000262(first column), A084357(row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> simplify(hypergeom([-n,-n-1],[],1)), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, Sum[BellY[n+1, k, Range[n+1]!], {k, 0, n+1}]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny_ *)

E.g.f.: exp(y*(exp(x/(1-x))-1)).

A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows.

Original entry on oeis.org

-2, 3, 3, -7, -12, -7, 25, 48, 48, 25, -121, -260, -240, -260, -121, 721, 1830, 1500, 1500, 1830, 721, -5041, -15162, -13230, -8400, -13230, -15162, -5041, 40321, 141176, 142296, 70560, 70560, 142296, 141176, 40321, -362881, -1451592, -1695456, -874944, -423360, -874944, -1695456, -1451592, -362881

Offset: 1

Roger L. Bagula, Apr 05 2010

			Triangle begins as:
     -2;
      3,      3;
     -7,    -12,     -7;
     25,     48,     48,    25;
   -121,   -260,   -240,  -260,   -121;
    721,   1830,   1500,  1500,   1830,    721;
  -5041, -15162, -13230, -8400, -13230, -15162,  -5041;
  40321, 141176, 142296, 70560,  70560, 142296, 141176, 40321;

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000262, A001263, A008297, A105278, A169154.

A001263:= func< n,k | Binomial(n-1, k-1)*Binomial(n,k-1)/k >;
A169653:= func< n,k | (-1)^n*A001263(n, k)*(Factorial(k) + Factorial(n-k+1)) >;
[A169653(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021

t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
T[n_, m_] = t[n, m] + t[n, n-m+1];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Feb 23 2021 *)

def A001263(n, k): return binomial(n-1, k-1)*binomial(n,k-1)/k
def A169653(n, k): return (-1)^n*A001263(n, k)*(factorial(k) + factorial(n-k+1))
flatten([[A169653(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Feb 23 2021

T(n, k) = t(n, k) + t(n, n-k+1), where t(n, k) = (-1)^n*(n!/k!)*binomial(n-1, k-1).
T(n, k) = A008297(n,k) + A008297(n,n-k+1).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1)).
T(n, k) = (-1)^n * ( k! + (n-k+1)! ) * A001263(n, k).
Sum_{k=1..n} T(n, k) = 2 * (-1)^n * A000262(n). (End)

Edited by G. C. Greubel, Feb 23 2021

A079005 Exponential transform of unsigned Lah-triangle |A008297(n,k)|.

Original entry on oeis.org

1, 2, 2, 6, 12, 5, 24, 72, 60, 15, 120, 480, 600, 300, 52, 720, 3600, 6000, 4500, 1560, 203, 5040, 30240, 63000, 63000, 32760, 8526, 877, 40320, 282240, 705600, 882000, 611520, 238728, 49112, 4140, 362880, 2903040, 8467200, 12700800, 11007360

Offset: 1

Vladeta Jovovic, Feb 01 2003

			1; 2,2; 6,12,5; 24,72,60,15; 120,480,600,300,52; ...

N. J. A. Sloane, Transforms

Cf. A055924, A055896, A056860.

E.g.f.: exp(exp(x*y/(1-x))-1). a(n, k) = n!/k! * binomial(n-1, k-1) * bell(k) = |A008297(n, k)| * A000110(k).

A079621 Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).

Original entry on oeis.org

1, 4, 1, 24, 12, 1, 192, 144, 24, 1, 1920, 1920, 480, 40, 1, 23040, 28800, 9600, 1200, 60, 1, 322560, 483840, 201600, 33600, 2520, 84, 1, 5160960, 9031680, 4515840, 940800, 94080, 4704, 112, 1, 92897280, 185794560, 108380160, 27095040, 3386880, 225792

Offset: 1

Vladeta Jovovic, Jan 29 2003

Also the Bell transform of A002866(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -1 <= d <= 2). - Shuhei Tsujie, Apr 26 2019

			Triangle begins:
     1;
     4,    1;
    24,   12,   1;
   192,  144,  24,  1;
  1920, 1920, 480, 40, 1;
  ...

P. Bala, The white diamond product of power series.
N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A002866 (first column), A025168 (row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 2^n*(n+1)!, 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2^#*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

E.g.f.: exp(x*y/(1-2*x)).
T(n, k) = n!/k!*binomial(n-1, k-1)*2^(n-k). - Vladeta Jovovic, Sep 24 2003
The n-th row polynomial equals x o (x + 2) o (x + 4) o ... o (x + 2*n), where o is the deformed Hadamard product of power series defined in Bala, section 3.1. - Peter Bala, Jan 18 2018

A156786 The triangular sequence of symmetrical Lah numbers (A111596, A008297) : L(n, m) = (-1)^n* binomial(n,k)binomial(n-1, k-1)( (n-k)! + (n-k)*(k-1)! ), with L(0,0) = 2, L(n,0) = L(n,n) = (-1)^n.

Original entry on oeis.org

2, -1, -1, 1, 4, 1, -1, -12, -12, -1, 1, 36, 72, 36, 1, -1, -140, -360, -360, -140, -1, 1, 750, 2100, 2400, 2100, 750, 1, -1, -5082, -15750, -16800, -16800, -15750, -5082, -1, 1, 40376, 142296, 152880, 117600, 152880, 142296, 40376, 1, -1, -362952, -1453536

Offset: 0

Roger L. Bagula, Feb 15 2009

Row sums are: {2, -2, 6, -26, 146, -1002, 8102, -75266, 788706, -9193106, 117882182, ...} = signed version of 2*A000262.

			Triangle begins as:
   2;
  -1,    -1;
   1,     4,      1;
  -1,   -12,    -12,     -1;
   1,    36,     72,     36,      1;
  -1,  -140,   -360,   -360,   -140,     -1;
   1,   750,   2100,   2400,   2100,    750,      1;
  -1, -5082, -15750, -16800, -16800, -15750,  -5082,    -1;
   1, 40376, 142296, 152880, 117600, 152880, 142296, 40376, 1;

J. Riordan, Combinatorial Identities, Wiley, 1968, p.48

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A111596, A008297.

[[n eq 0 and k eq 0 select 2 else k eq 0 or k eq n select (-1)^n else (-1)^n*Binomial(n,k)*Binomial(n-1, k-1)*( Factorial(n-k) + (n-k)* Factorial(k-1) ): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 20 2019

L[n_, k_]:= If[n==0 && k==0, 2, If[k==0 || k==n, (-1)^n, (-1)^n* Binomial[n,k]*Binomial[n-1,k-1]*( (n-k)! + (n-k)*(k-1)! )]]; Table[L[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

{ L(n, k) = if(n==0 && k==0, 2, if(k==0 || k==n, (-1)^n, (-1)^n* binomial(n,k)*binomial(n-1, k-1)*( (n-k)! + (n-k)*(k-1)! ) )) }; \\ G. C. Greubel, May 20 2019

def L(n, k):
    if (k==0 and n==0): return 2
    elif (k==0 or k==n): return (-1)^n
    else: return (-1)^n*binomial(n,k)*binomial(n-1, k-1)*( factorial(n-k) + (n-k)*factorial(k-1) )
[[L(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 20 2019

L(n, m) = if m = 0 then KroneckerDelta(n, 0) otherwise (-1)^n*(n!/m!)* binomial(n-1, m-1) + if m = n then KroneckerDelta(n, 0) otherwise (-1)^n* n! *binomial(n,m)* binomial(n-1, n-m-1).

L(n, m) = (-1)^n* binomial(n,k)*binomial(n-1, k-1)*( (n-k)! + (n-k)*(k-1)! ), with L(0,0) = 2, L(n,0) = L(n,n) = (-1)^n. - G. C. Greubel, May 20 2019

Edited by G. C. Greubel, May 20 2019

A223512 Triangle T(n,k) represents the coefficients of (x^10*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, 10, 1, 190, 30, 1, 5320, 1060, 60, 1, 196840, 45600, 3400, 100, 1, 9054640, 2340040, 208800, 8300, 150, 1, 498005200, 140096880, 14241640, 690200, 17150, 210, 1, 31872332800, 9604302400, 1080045120, 60485040, 1856400, 31640, 280, 1, 2326680294400

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
10,1;
190,30,1;
5320,1060,60,1;
196840,45600,3400,100,1;
9054640,2340040,208800,8300,150,1;
498005200,140096880,14241640,690200,17150,210,1;
31872332800,9604302400,1080045120,60485040,1856400,31640,280,1,2326680294400

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

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

A223514 Triangle T(n,k) represents the coefficients of (x^12*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, 12, 1, 276, 36, 1, 9384, 1536, 72, 1, 422280, 80040, 4920, 120, 1, 23647680, 4984560, 365400, 12000, 180, 1, 1584394560, 362597760, 30197160, 1205400, 24780, 252, 1, 123582775680, 30229617600, 2778370560, 127834560, 3237360, 45696, 336, 1, 1099867035520

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
12,1;
276,36,1;
9384,1536,72,1;
422280,80040,4920,120,1;
23647680,4984560,365400,12000,180,1;
1584394560,362597760,30197160,1205400,24780,252,1;
123582775680,30229617600,2778370560,127834560,3237360,45696,336,1;
1099867035520,...

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^12*diff(b[j-1],x$1);
end do;

A223515 Triangle T(n,k) represents the coefficients of (x^13*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, 13, 1, 325, 39, 1, 12025, 1807, 78, 1, 589225, 102375, 5785, 130, 1, 35942725, 6936475, 466830, 14105, 195, 1, 2623818925, 549241875, 41948725, 1538810, 29120, 273, 1, 223024608625, 49858620175, 4198780950, 177364005, 4130490, 53690, 364, 1, 21633387036625

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
13,1;
325,39,1;
12025,1807,78,1;
589225,102375,5785,130,1;
35942725,6936475,466830,14105,195,1
2623818925,549241875,41948725,1538810,29120,273,1;
223024608625,49858620175,4198780950,177364005,4130490,53690,364,1;
21633387036625,...

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^13*diff(b[j-1],x$1);
end do;

cp's OEIS Frontend

A079639 Matrix product of Stirling1-triangle A008275(n,k) and unsigned Lah-triangle |A008297(n,k)|.

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A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

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A088814 Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling2-triangle A008277(n,k).

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A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows.

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A079005 Exponential transform of unsigned Lah-triangle |A008297(n,k)|.

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A079621 Matrix square of unsigned Lah triangle abs(A008297(n,k)) or A105278(n,k).

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A156786 The triangular sequence of symmetrical Lah numbers (A111596, A008297) : L(n, m) = (-1)^n* binomial(n,k)*binomial(n-1, k-1)*( (n-k)! + (n-k)*(k-1)! ), with L(0,0) = 2, L(n,0) = L(n,n) = (-1)^n.

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A156786 The triangular sequence of symmetrical Lah numbers (A111596, A008297) : L(n, m) = (-1)^n* binomial(n,k)binomial(n-1, k-1)( (n-k)! + (n-k)*(k-1)! ), with L(0,0) = 2, L(n,0) = L(n,n) = (-1)^n.