A008297 -id:A008297 - cp's OEIS frontend

A176022 Triangle T(n, k) = A176013(n, k) + A176013(n, n-k+1), read by rows.

-2, 3, 3, -7, -18, -7, 25, 96, 96, 25, -121, -650, -800, -650, -121, 721, 5490, 7500, 7500, 5490, 721, -5041, -53067, -92610, -73500, -92610, -53067, -5041, 40321, 564704, 1328096, 987840, 987840, 1328096, 564704, 40321, -362881, -6532164, -20345472, -18373824, -10668672, -18373824, -20345472, -6532164, -362881

Offset: 1

Roger L. Bagula, Apr 06 2010

			Triangle begins as:
     -2;
      3,      3;
     -7,    -18,      -7;
     25,     96,      96,     25;
   -121,   -650,    -800,   -650,   -121;
    721,   5490,    7500,   7500,   5490,     721;
  -5041, -53067,  -92610, -73500, -92610,  -53067,  -5041;
  40321, 564704, 1328096, 987840, 987840, 1328096, 564704, 40321;

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

Cf. A008297, A176013, A176021.

A176013:= func< n, k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;
[A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 15 2021

(* First program *)
T[n_, m_]:= ((-1)^n*n!/(m*m!))*Binomial[n-1, m-1]*Binomial[n, m-1] + ((-1)^n*n!)/((n-m+1)*(n-m+1)!)*Binomial[n-1, n-m] Binomial[n, n-m];
Table[T[n, m], {n,10}, {m,n}]//Flatten
(* Second program *)
A176013[n_, k_] := (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];
T[n_, k_]:= A176013[n, k] + A176013[n, n-k+1];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)

def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)
flatten([[A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 15 2021

T(n, k) = ((-1)^n*n!/(k*k!))*binomial(n-1, k-1)*binomial(n, k-1) + ((-1)^n*n!)/((n-k+1)*(n-k+1)!)*binomial(n-1, n-k)*binomial(n, n-k).
From G. C. Greubel, Feb 15 2021: (Start)
T(n, k) = A176013(n, k) + A176013(n, n-k+1), where A176013(n, k) = (-1)^n*(n!/(k*k!))*binomial(n-1, k-1)*binomial(n, k-1).
Sum_{k=1..n} T(n, k) = 2*(-1)^n * n! * Hypergeometric2F2(-n, -(n-1); 2, 2; 1). (End)

Edited by G. C. Greubel, Feb 15 2021

A317364 Expansion of e.g.f. exp(2*x/(1 + x)).

Original entry on oeis.org

1, 2, 0, -4, 16, -48, 64, 800, -12288, 127232, -1150976, 9266688, -58726400, 68777984, 7510646784, -207794409472, 4241007640576, -77359570944000, 1321952191971328, -21274345818161152, 313768799799607296, -3838962981483839488, 21775623343518515200, 859024717017756205056

Offset: 0

Ilya Gutkovskiy, Jul 26 2018

sign

Inverse Lah transform of the powers of 2 (A000079).

Robert Israel, Table of n, a(n) for n = 0..451
N. J. A. Sloane, Transforms

Cf. A000079, A052897, A111884.

Cf. A001263, A008297, A086915.

[n eq 0 select 1 else (-1)^n*Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), 2): n in [0..25]]; // G. C. Greubel, Feb 23 2021

a:= proc(n) option remember; add((-1)^(n-k)*
      n!/k!*binomial(n-1, k-1)*2^k, k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

nmax = 23; CoefficientList[Series[Exp[2 x/(1 + x)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[Exp[-2 (-x)^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] 2^k n!/k!, {k, 0, n}], {n, 0, 23}]
Join[{1}, Table[2 (-1)^(n+1) n! Hypergeometric1F1[1-n, 2, 2], {n, 23}]]

a(n) = if (n==0, 1, (-1)^n*n!*pollaguerre(n, -1, 2)); \\ Michel Marcus, Feb 23 2021

[1 if n==0 else (-1)^n*factorial(n)*gen_laguerre(n, -1, 2) for n in (0..25)] # G. C. Greubel, Feb 23 2021

E.g.f.: Product_{k>=1} exp(-2*(-x)^k).
a(n) = 2*(-1)^(n+1) * n! * Hypergeometric1F1([1-n], [2], 2).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*2^k*n!/k!.
(n^2 + n)*a(n) + 2*n*a(n+1) + a(n+2) = 0. - Robert Israel, Aug 18 2019
From G. C. Greubel, Feb 23 2021: (Start)
a(n) = (-1)^n * n! * Laguerre(n, -1, 2) for n > 0 with a(0) = 1.
a(n) = Sum_{k=0..n} (-1)^(n-k) * A086915(n, k).
a(n) = (-1)^n * Sum_{k=0..n} 2^k * A008297(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)! * A001263(n, k). (End)

A059374 Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

Original entry on oeis.org

1, 3, 2, 13, 18, 6, 73, 156, 108, 24, 501, 1460, 1560, 720, 120, 4051, 15030, 21900, 15600, 5400, 720, 37633, 170142, 315630, 306600, 163800, 45360, 5040, 394353, 2107448, 4763976, 5891760, 4292400, 1834560, 423360, 40320

Offset: 1

Vladeta Jovovic, Jan 28 2001

L'(n, i) are unsigned Lah numbers (Cf. A008297).

			Triangle begins:
  [1],
  [3, 2],
  [13, 18, 6],
  [73, 156, 108, 24],
  [501, 1460, 1560, 720, 120],
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. T(n, 0) = A000262, A025168 (row sums), A000012 (alternating row sums), A059110.

Cf. A008297, A271703.

t[n_, k_] := Sum[ Binomial[n-1, n-i-1]*n!/(n-i)!*Binomial[i, k], {i, 0, n}]; Table[t[n, k], {n, 1, 8}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Mar 22 2013 *)

for(n=1,10, for(k=0,n-1, print1(sum(j=0,n, binomial(j,k)* binomial(n-1,n-j-1)*n!/(n-j)!), ", "))) \\ G. C. Greubel, Jan 29 2018

E.g.f.: exp(x/(1-(1+y)*x))/(1-(1+y)*x)^2. - Vladeta Jovovic, May 10 2003

A111599 Lah numbers: a(n) = n!*binomial(n-1,8)/9!.

Original entry on oeis.org

1, 90, 4950, 217800, 8494200, 309188880, 10821610800, 371026656000, 12614906304000, 428906814336000, 14668613050291200, 506733905373696000, 17735686688079360000, 630299019222512640000, 22780807409042242560000

Offset: 9

Wolfdieter Lang, Aug 23 2005

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

Column 9 of unsigned A008297 and A111596.
Column 8: A111598.
Cf. A001113, A001620, A091725, A099285.

part_ZL:=[S,{S=Set(U,card=r),U=Sequence(Z,card>=1)}, labeled]: seq(count(subs(r=9,part_ZL),size=m),m=9..23) ; # Zerinvary Lajos, Mar 09 2007

Table[n!*Binomial[n-1, 8]/9!, {n, 9, 30}] (* Wesley Ivan Hurt, Dec 10 2013 *)

E.g.f.: ((x/(1-x))^9)/9!.
a(n) = (n!/9!)*binomial(n-1, 9-1).
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j), then a(n) = (-1)^(n-1)*f(n,9,-9), n >= 9. - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=9} 1/a(n) = 564552*(Ei(1) - gamma) - 264528*e - 873657/35, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=9} (-1)^(n+1)/a(n) = 28393416*(gamma - Ei(-1)) - 16938720/e - 573537159/35, where Ei(-1) = -A099285. (End)

A111600 Lah numbers: a(n) = n!*binomial(n-1,9)/10!.

Original entry on oeis.org

1, 110, 7260, 377520, 17177160, 721440720, 28857628800, 1121325004800, 42890681433600, 1629845894476800, 61934143990118400, 2364758225077248000, 91043191665474048000, 3543681152517682176000, 139722285442125754368000

Offset: 10

Wolfdieter Lang, Aug 23 2005

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.

Column 10 of unsigned A008297 and A111596.
Column 9: A111599.
Cf. A001113, A001620, A091725, A099285.

Table[n! * Binomial[n - 1, 9]/10!, {n, 10, 25}] (* Amiram Eldar, May 02 2022 *)

E.g.f.: ((x/(1-x))^10)/10!.
a(n) = (n!/10!)*binomial(n-1, 10-1).
If we define f(n,i,x) = Sum_{k=1..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i)*x^(k-j) then a(n) = (-1)^n*f(n,10,-10), (n>=10). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=10} 1/a(n) = 5086710*(gamma - Ei(1)) + 50940*e + 91914449/14, where gamma = A001620, Ei(1) = A091725 and e = A001113.
Sum_{n>=10} (-1)^n/a(n) = 413689770*(gamma - Ei(-1)) - 246749400/e - 3342795017/14, where Ei(-1) = -A099285. (End)

A169654 Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -4, 1, 1, 24, 24, 1, 1, -138, -118, -138, 1, 1, 1110, 780, 780, 1110, 1, 1, -10120, -8188, -3358, -8188, -10120, 1, 1, 100856, 101976, 30240, 30240, 101976, 100856, 1, 1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710, 1

Offset: 1

Roger L. Bagula, Apr 05 2010

			Triangle begins as:
  1;
  1,        1;
  1,       -4,        1;
  1,       24,       24,       1;
  1,     -138,     -118,    -138,      1;
  1,     1110,      780,     780,   1110,       1;
  1,   -10120,    -8188,   -3358,  -8188,  -10120,        1;
  1,   100856,   101976,   30240,  30240,  101976,   100856,        1;
  1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710,        1;
  1, 12700890, 18147240, 9132480, 816480,  816480,  9132480, 18147240, 12700890, 1;

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

Cf. A000262, A001263, A008297, A105278, A169653.

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)) >;
A169654:= func< n,k | A169653(n, k) - A169653(n, 1) + 1 >;
[A169654(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] - (-1)^n*(n! + 1) + 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))
def A169654(n, k): return A169653(n, k) - A169653(n, 1) + 1
flatten([[A169654(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) - t(n, 1) - t(n, n) + 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) - (A008297(n,1) + A008297(n,n)) + 1.
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = A169653(n, k) - A169653(n, 1) + 1
T(n, k) = A169653(n, k) - (-1)^n * (n! + 1) + 1.
T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1) - (n! + 1) + (-1)^n).
Sum_{k=1..n} T(n, k) = (-1)^n *(2 * A000262(n) - n*(n! + 1) + (-1)^n * n). (End)

Edited by G. C. Greubel, Feb 23 2021

A169656 Triangle, read by rows, T(n, k) = (-1)^n(n!/k!)^2binomial(n-1, k-1).

Original entry on oeis.org

-1, 4, 1, -36, -18, -1, 576, 432, 48, 1, -14400, -14400, -2400, -100, -1, 518400, 648000, 144000, 9000, 180, 1, -25401600, -38102400, -10584000, -882000, -26460, -294, -1, 1625702400, 2844979200, 948326400, 98784000, 3951360, 65856, 448, 1

Offset: 1

Roger L. Bagula, Apr 05 2010

Row sums are: {-1, 5, -55, 1057, -31301, 1319581, -74996755, 5521809665, -510921831817, 58003632177301, ...}.

			Triangle begins as:
         -1;
          4,         1;
        -36,       -18,        -1;
        576,       432,        48,       1;
     -14400,    -14400,     -2400,    -100,     -1;
     518400,    648000,    144000,    9000,    180,    1;
  -25401600, -38102400, -10584000, -882000, -26460, -294, -1;

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

Cf. A008297.

F:=Factorial;; Flat(List([1..10], n-> List([1..n], k-> (-1)^n*(F(n)/F(k) )^2*Binomial(n-1, k-1) ))); # G. C. Greubel, Nov 28 2019

F:=Factorial; [(-1)^n*(F(n)/F(k))^2*Binomial(n-1, k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 28 2019

seq(seq( (-1)^n*(n!/k!)^2*binomial(n-1, k-1), k=1..n), n=1..10); # G. C. Greubel, Nov 28 2019

T[n_, k_]:= (-1)^n*(n!/k!)^2*Binomial[n-1, k-1]; Table[T[n, k], {n,10}, {k,n}]//Flatten

T(n,k) = (-1)^n*(n!/k!)^2*binomial(n-1, k-1); \\ G. C. Greubel, Nov 28 2019

f=factorial; [[(-1)^n*(f(n)/f(k))^2*binomial(n-1, k-1) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 28 2019

T(n, k) = (-1)^n * (n!/k!)^2 * binomial(n-1, k-1).

Edited by G. C. Greubel, Nov 28 2019

A169658 Triangle, read by rows, defined by T(n, k) = b(n, k) + b(n, n-k+1) - (b(n,1) + b(n,n)) + 1, where b(n, k) = (-1)^n(n!/k!)^2 binomial(n-1, k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, -96, -96, 1, 1, -98, 9602, -98, 1, 1, 129780, -365400, -365400, 129780, 1, 1, -12701092, 14791142, 23637602, 14791142, -12701092, 1, 1, 1219277248, -677310144, -1522967040, -1522967040, -677310144, 1219277248, 1

Offset: 1

Roger L. Bagula, Apr 05 2010

Row sums are: {1, 2, 4, -190, 9408, -471238, 27817704, -1961999870, 163293385984, -15674630045398, ...}.

			Triangle begins as:
  1;
  1,         1;
  1,         2,        1;
  1,       -96,      -96,        1;
  1,       -98,     9602,      -98,        1;
  1,    129780,  -365400,  -365400,   129780,         1;
  1, -12701092, 14791142, 23637602, 14791142, -12701092, 1;

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

Cf. A008297.

b:= func< n,k | (-1)^n*(Factorial(n)/Factorial(k))^2*Binomial(n-1, k-1) >;
[[b(n, k) +b(n, n-k+1) -b(n,1) -b(n,n) +1: k in [1..n]]: n in [1..10]]; // G. C. Greubel, May 20 2019

L[n_, m_] = (-1)^n*(n!/m!)^2*Binomial[n-1, m-1];
t[n_, m_] = L[n, m] + L[n, n-m+1];
Table[t[n, m] - t[n, 1] + 1, {n, 1, 10}, {m, 1, n}]//Flatten

b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1);
t(n, k) = b(n, k) + b(n, n-k+1);
for(n=1, 10, for(k=1, n, print1(t(n,k) - t(n,1) + 1, ", "))) \\ G. C. Greubel, May 20 2019

def b(n, k): return (-1)^n*factorial(n-k)^2*binomial(n,k)^2*binomial(n-1, k-1)
def t(n, k): return b(n, k) + b(n, n-k+1)
[[t(n,k) - t(n,1) + 1 for k in (1..n)] for n in (1..10)] # G. C. Greubel, May 20 2019

T(n, k) = b(n, k) + b(n, n-k+1) - b(n, n) - b(n, 1) + 1, where b(n, k) = (-1)^n*(n!/m!)^2 *binomial(n-1, k-1), where 1 <= k <= n, n >= 1.

Edited by G. C. Greubel, May 20 2019

A173227 Partial sums of A000262.

Original entry on oeis.org

1, 2, 5, 18, 91, 592, 4643, 42276, 436629, 5033182, 63974273, 888047414, 13358209647, 216334610860, 3751352135263, 69325155322184, 1359759373992105, 28206375825238458, 616839844140642301, 14181213537729200474, 341879141423814854915, 8623032181189674581256

Offset: 0

Jonathan Vos Post, Feb 13 2010

nonn

Partial sums of the number of "sets of lists": number of partitions of {1,..,n} into any number of lists, where a list means an ordered subset. The subsequence of primes begins: 2, 5, 4643, 616839844140642301.

			a(20) = 1 + 1 + 3 + 13 + 73 + 501 + 4051 + 37633 + 394353 + 4596553 + 58941091 + 824073141 + 12470162233 + 202976401213 + 3535017524403 + 65573803186921 + 1290434218669921 + 26846616451246353 + 588633468315403843 + 13564373693588558173 + 327697927886085654441.

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A001263, A001700, A002868, A008297, A052852, A066668, A111596,.

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[n eq 0 select 1 else 1 + (&+[ Factorial(j)*( l(j,-1) - l(j-1,-1) ): j in [1..n]]): n in [0..25]]; // G. C. Greubel, Mar 09 2021

b:= proc(n) option remember; `if`(n=0, 1, add(
       b(n-j)*j!*binomial(n-1, j-1), j=1..n))
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

With[{m = 25}, CoefficientList[Exp[x/(1-x)] + O[x]^m, x] Range[0, m-1]!// Accumulate] (* Jean-François Alcover, Nov 21 2020 *)
Table[1 +Sum[j!*(LaguerreL[j, -1] -LaguerreL[j-1, -1]), {j,n}], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[1 + sum(factorial(j)*(gen_laguerre(j,0,-1) - gen_laguerre(j-1,0,-1)) for j in (1..n)) for n in (0..30)] # G. C. Greubel, Mar 09 2021

From Vaclav Kotesovec, Oct 25 2016: (Start)
a(n) = 2*n*a(n-1) - (n^2 - n + 1)*a(n-2) + (n-2)*(n-1)*a(n-3).
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/sqrt(2) * (1 - 5/(48*sqrt(n))).
(End)
a(n) = 1 + Sum_{j=1..n} j!*( LaguerreL(j,-1) - LaguerreL(j-1,-1) ). - G. C. Greubel, Mar 09 2021

A276960 a(n) = A000262(n)^2.

Original entry on oeis.org

1, 1, 9, 169, 5329, 251001, 16410601, 1416242689, 155514288609, 21128299481809, 3474052208270281, 679096541717605881, 155504946117339546289, 41199419449380747871369, 12496348897836314700506409

Offset: 0

Emanuele Munarini, Sep 27 2016

nonn

Cf. A000262, A105278, A008297, A276964.

Table[HypergeometricPFQ[{-n+1,-n},{},1]^2,{n,0,100}]

makelist(hypergeometric([-n+1,-n],[],1)^2,n,0,12);

Recurrence: (2*n+3)*a(n+3)-(2*n+5)*(3*n^2+13*n+13)*a(n+2)+(n+2)*(n+1)*(2*n+3)*(3*n^2+13*n+13)*a(n+1)-n^2*(n+1)^3*(n+2)*(2 n+5)*a(n) = 0.

Asymptotic: a(n) ~ exp(-2*n+4*sqrt(n)-1)*n^(2*n-1/2)/2 * (1 - 5/(24*sqrt(n)) - 35/(1152*n)).

cp's OEIS Frontend

A176022 Triangle T(n, k) = A176013(n, k) + A176013(n, n-k+1), read by rows.

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A317364 Expansion of e.g.f. exp(2*x/(1 + x)).

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A059374 Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

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A111599 Lah numbers: a(n) = n!*binomial(n-1,8)/9!.

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A111600 Lah numbers: a(n) = n!*binomial(n-1,9)/10!.

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A169654 Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.

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A169656 Triangle, read by rows, T(n, k) = (-1)^n*(n!/k!)^2*binomial(n-1, k-1).

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A169656 Triangle, read by rows, T(n, k) = (-1)^n(n!/k!)^2binomial(n-1, k-1).

A169658 Triangle, read by rows, defined by T(n, k) = b(n, k) + b(n, n-k+1) - (b(n,1) + b(n,n)) + 1, where b(n, k) = (-1)^n(n!/k!)^2 binomial(n-1, k-1).