A007318 -id:A007318 - cp's OEIS frontend

A132750 A132749 * [1, 2, 3, ...] = A007318 * A065190.

1, 4, 9, 21, 49, 113, 257, 577, 1281, 2817, 6145, 13313, 28673, 61441, 131073, 278529, 589825, 1245185, 2621441, 5505025, 11534337, 24117249, 50331649, 104857601, 218103809, 452984833, 939524097, 1946157057, 4026531841

Offset: 0

Gary W. Adamson, Aug 28 2007

Equals double binomial transform of [1, 2, -3, 7, -15, 31, -63, 127, -255, ...]. - Gary W. Adamson, Jul 23 2008

For n >= 1, also the number of cliques in the n-hypercube graph Q_n. - Eric W. Weisstein, Mar 31 2017

			a(3) = 21 = (1, 3, 3, 1) dot (1, 3, 2, 5) = (1 + 9 + 6 + 5) = 21; where A065190 = (1, 3, 2, 5, 4, 7, 6, 9, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Jelinek, T. Mansour, M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, Lemma 4.21 (sequence starting 1, 1, 2, 4, 9, 21, .... with offset 0).
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Hypercube Graph
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A065190, A132749, A135224.

Concatenation([1], List([1..30], n-> n*2^(n-1) + 2^n + 1 )); # G. C. Greubel, Nov 20 2019

[n*2^(n-1) + 2^n + 1 - 0^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 26 2014

A132750:=n->n*2^(n-1)+2^n+1-0^n: seq(A132750(n), n=0..30); # Wesley Ivan Hurt, Sep 26 2014

Join[{1}, Table[n*2^(n-1) +2^n +1, {n, 30}]] (* Wesley Ivan Hurt, Sep 26 2014 *)
Join[{1}, LinearRecurrence[{5,-8,4}, {4,9,21}, 30]] (* Vincenzo Librandi, Apr 01 2017 *)

vector(31, n, if(n==1, 1, (n-1)*2^(n-2) + 2^(n-1) + 1)) \\ G. C. Greubel, Nov 20 2019

[1]+[n*2^(n-1) + 2^n + 1 for n in (1..30)] # G. C. Greubel, Nov 20 2019

A132749 as an infinite lower triangular matrix * vector [1, 2, 3, ...]. Binomial transform of A065190 (with an incorrect offset)
Row sums of triangle A135224. - Gary W. Adamson, Nov 23 2007
G.f.: (1-x-3*x^2+4*x^3)/((1-x)*(1-2*x)^2). - Colin Barker, Aug 09 2012
a(n) = n*2^(n-1) + 2^n + 1 - 0^n. - Tim Smith, Sep 25 2014
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3). - Wesley Ivan Hurt, Sep 26 2014
E.g.f.: -1 + exp(x) + (1+x)*exp(2*x). - G. C. Greubel, Nov 20 2019

A132823 A007318 + 2A103451 - 2A000012.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 8, 8, 3, 1, 1, 4, 13, 18, 13, 4, 1, 1, 5, 19, 33, 33, 19, 5, 1, 1, 6, 26, 54, 68, 54, 26, 6, 1, 1, 7, 34, 82, 124, 124, 82, 34, 7, 1, 1, 8, 43, 118, 208, 250, 208, 118, 43, 8, 1, 1, 9, 53, 163, 328, 460, 460, 328, 163, 53, 9, 1

Offset: 0

Gary W. Adamson, Sep 02 2007

Row sums = A132824: (1, 2, 2, 4, 10, 24, 54, 116, 242, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 0,  1;
  1, 1,  1,  1;
  1, 2,  4,  2,   1;
  1, 3,  8,  8,   3,   1;
  1, 4, 13, 18,  13,   4,  1;
  1, 5, 19, 33,  33,  19,  5,  1;
  1, 6, 26, 54,  68,  54, 26,  6, 1;
  1, 7, 34, 82, 124, 124, 82, 34, 7, 1;
  ...

Cf. A007318, A103451, A000012, A132824.

A(2n,n) gives A115112 for n>0.

A007318 + 2*A103451 - 2*A000012 as infinite lower triangular matrices.

One missing 1 inserted and more terms added by Alois P. Heinz, Feb 10 2019

A133084 A007318 * A133080.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 4, 1, 5, 4, 10, 4, 1, 6, 5, 20, 10, 6, 1, 7, 6, 35, 20, 21, 6, 1, 8, 7, 56, 35, 56, 21, 8, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 10, 9, 120, 84, 252, 126, 120, 36, 10, 1, 11, 10, 165, 120, 462, 252, 330, 120, 55, 10, 1

Offset: 1

Gary W. Adamson, Sep 16 2007

Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...).

A133084 is jointly generated with A133567 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012

			First few rows of the triangle:
  1;
  2,  1;
  3,  2,  1;
  4,  3,  4,  1;
  5,  4, 10,  4,  1;
  6,  5, 20, 10,  6,  1;
  7,  6, 35, 20, 21,  6,  1;
  ...

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

Cf. A133080, A003945, A133567.

Cf. A000292 (column 3 and 4), A000389 (column 5 and 6), A000580 (column 7).

/* As triangle */ [[(1-(1+(-1)^k)/2 )*Binomial(n, k)+((1+(-1)^k)/2)*Binomial(n-1, k-1): k in [1..n]]: n in [1.. 11]]; // Vincenzo Librandi, Oct 21 2017

A133084 := proc(n,k)
    add(binomial(n-1,i-1)*A133080(i,k),i=1..n) ;
end proc: # R. J. Mathar, Jun 13 2025

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A133567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A133084 *)
(* Clark Kimberling, Feb 28 2012 *)
T[n_, k_] := If[k == n, 1, (1  - (1 + (-1)^k)/2 )*Binomial[n, k] + ((1 + (-1)^k)/2)*Binomial[n - 1, k - 1]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

for(n=1,10, for(k=1,n, print1(if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)), ", "))) \\ G. C. Greubel, Oct 21 2017

Binomial transform of triangle A133080.

A133124 A007318 * [1, 2, 2, 3, 2, 3, 2, 3, 2, ...].

Original entry on oeis.org

1, 3, 7, 16, 35, 74, 153, 312, 631, 1270, 2549, 5108, 10227, 20466, 40945, 81904, 163823, 327662, 655341, 1310700, 2621419, 5242858, 10485737, 20971496, 41943015, 83886054, 167772133, 335544292, 671088611, 1342177250, 2684354529, 5368709088, 10737418207, 21474836446

Offset: 0

Gary W. Adamson, Sep 19 2007

			a(3) = (1, 3, 3, 1) dot (1, 2, 2, 3) = (1 + 6 + 6 + 3).
a(5) = 74 = 2^6 + 2^4 - 6 = 64 + 16 - 6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000079, A133938.

[1] cat [5*2^(n - 1) - n -1: n in [1..50]]; // G. C. Greubel, Oct 21 2017

Join[{1}, Table[5*2^(n-1) - n -1, {n,1,50}]] (* G. C. Greubel, Oct 21 2017 *)
LinearRecurrence[{4,-5,2},{1,3,7,16},40] (* Harvey P. Dale, Jun 18 2024 *)

concat(1, for(n=1,50, print1(5*2^(n - 1) - n - 1, ", "))) \\ G. C. Greubel, Oct 21 2017

Binomial transform of [1, 2, 2, 3, 2, 3, 2, 3, 2, ...].
a(n) = 5*2^(n-1) - (n+1), for n >= 1.
Row sums of triangle A133938. - Gary W. Adamson, Sep 29 2007
G.f.: 1 + x*(3-5*x+3*x^2)/((1-2*x)*(1-x)^2). - R. J. Mathar, Nov 14 2007
E.g.f.: (5*exp(2*x) - 2*(1+x)*exp(x) - 1)/2. - G. C. Greubel, Oct 21 2017

Terms a(9) onward added by G. C. Greubel, Oct 21 2017

A134083 A007318 * A134082.

Original entry on oeis.org

1, 3, 1, 5, 6, 1, 7, 15, 9, 1, 9, 28, 30, 12, 1, 11, 45, 70, 50, 15, 1, 13, 66, 135, 140, 75, 18, 1, 15, 91, 231, 315, 245, 105, 21, 1, 17, 120, 364, 616, 630, 392, 140, 24, 1, 19, 153, 540, 1092, 1386, 1134, 588, 180, 27, 1

Offset: 0

Gary W. Adamson, Oct 07 2007

Row sums = A001787: (1, 4, 12, 32, 80, 192, ...).

A134083 * [1,2,3,...] = A084850: (1, 5, 20, 68, 208, 592, ...).

			First few rows of the triangle:
   1;
   3,  1;
   5,  6,   1;
   7, 15,   9,   1;
   9, 28,  30,  12,   1;
  11, 45,  70,  50,  15,   1;
  13, 66, 135, 140,  75,  18,  1;
  15, 91, 231, 315, 245, 105, 21, 1;
  ...

Cf. A134082, A001787, A084850.

Binomial transform of A134082
From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] exp(x)(1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e., I_o[2*(u*x)^(1/2)] = Sum_{j>=0} u^j/j! * x^j/j!. - Tom Copeland, Dec 07 2007
Row polynomial e.g.f.: exp(x*y) * exp(x) * (1+2x). - Tom Copeland, Dec 03 2013

A135837 A007318 * a triangle with (1, 2, 2, 4, 4, 8, 8, ...) in the main diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 6, 6, 4, 1, 8, 12, 16, 4, 1, 10, 20, 40, 20, 8, 1, 12, 30, 80, 60, 48, 8, 1, 14, 42, 140, 140, 168, 56, 16, 1, 16, 56, 224, 280, 448, 224, 128, 16, 1, 18, 72, 336, 504, 1008, 672, 576, 144, 32

Offset: 1

Gary W. Adamson, Dec 01 2007

This sequence is jointly generated with A117919 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 26 2012

Subtriangle of the triangle (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012

			First few rows of the triangle:
  1;
  1,  2;
  1,  4,  2;
  1,  6,  6,  4;
  1,  8, 12, 16,  4;
  1, 10, 20, 40, 20,  8;
  1, 12, 30, 80, 60, 48,  8;
  ...
From _Philippe Deléham_, Mar 19 2012: (Start)
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, -1, -1, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  4,  2,  0;
  1,  6,  6,  4,  0;
  1,  8, 12, 16,  4,  0;
  1, 10, 20, 40, 20,  8,  0;
  1, 12, 30, 80, 60, 48, 8,  0; (End)

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A000217, A001333 (row sums), A007318, A016116, A117919, A135838.

a135837 n k = a135837_tabl !! (n-1) !! (k-1)
a135837_row n = a135837_tabl !! (n-1)
a135837_tabl = [1] : [1, 2] : f [1] [1, 2] where
   f xs ys = ys' : f ys ys' where
     ys' = zipWith3 (\u v w -> 2 * u - v + 2 * w)
                    (ys ++ [0]) (xs ++ [0, 0]) ([0, 0] ++ xs)
-- Reinhard Zumkeller, Aug 08 2012

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= 2 x*u[n-1, x] + v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A117919 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A135837 *) (* Clark Kimberling, Feb 26 2012 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1, 1, If[k==n, 2^Floor[n/2], 2*T[n-1, k] - T[n-2, k] + 2*T[n-2, k-2]]]];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Feb 07 2022 *)

def T(n,k): # A135837
    if (k<1 or k>n): return 0
    elif (k==1): return 1
    elif (k==n): return 2^(n//2)
    else: return 2*T(n-1, k) - T(n-2, k) + 2*T(n-2, k-2)
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 07 2022

Binomial transform of a triangle with (1, 2, 2, 4, 4, 8, 8, ...) in the main diagonal and the rest zeros.
Sum_{k=1..n} T(n, k) = A001333(n).
From Philippe Deléham, Mar 19 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-x+2*y*x^2-2*y^2*x^2)/(1-2*x+2*y*x^2-2*y^2*x^2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. (End)
G.f.: x*y*(1-x+2*x*y)/(1-2*x-2*x^2*y^2+x^2). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Feb 07 2022: (Start)
T(n, n) = A016116(n).
T(n, 2) = 2*(n-1).
T(n, 3) = 2*A000217(n-2). (End)

A143690 a(n) = A007318 * [1, 6, 14, 9, 0, 0, 0, ...].

Original entry on oeis.org

1, 7, 27, 70, 145, 261, 427, 652, 945, 1315, 1771, 2322, 2977, 3745, 4635, 5656, 6817, 8127, 9595, 11230, 13041, 15037, 17227, 19620, 22225, 25051, 28107, 31402, 34945, 38745, 42811, 47152, 51777, 56695, 61915, 67446, 73297, 79477, 85995, 92860, 100081, 107667

Offset: 0

Gary W. Adamson, Aug 29 2008

Binomial transform of [1, 6, 14, 9, 0, 0, 0,...].

Row sums of triangle A033292.

			a(3) = 70 = (1, 3, 3, 1) dot (1, 6, 14, 9) = (1 + 18 + 42 + 9). a(3) = 70 = sum of row 3 terms of triangle A033292: (13 + 16 + 19, + 22).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000326, A002412, A033292.

Cf. A226449. - Bruno Berselli, Jun 09 2013

Table[(n+1)*(3*n^2+2*n+2)/2, {n,0,50}] (* G. C. Greubel, May 30 2021 *)

[(n+1)*(3*n^2+2*n+2)/2 for n in (0..50)] # G. C. Greubel, May 30 2021

From R. J. Mathar, Aug 29 2008: (Start)
G.f.: (1 +3*x +5*x^2)/(1-x)^4.
a(n) = A002412(n+1) + 5*A000292(n-1). (End)
a(n) = A000326(n+1) + (n+1)*A000326(n). - Bruno Berselli, Jun 07 2013
From G. C. Greubel, May 30 2021: (Start)
a(n) = (n+1)*(3*n^2 +2*n +2)/2.
E.g.f.: (1/2)*(2 +12*x +14*x^2 +3*x^3)*exp(x). (End)

Extended beyond a(14) by R. J. Mathar, Aug 29 2008

A163866 Partial sums of A007318.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 11, 14, 15, 16, 20, 26, 30, 31, 32, 37, 47, 57, 62, 63, 64, 70, 85, 105, 120, 126, 127, 128, 135, 156, 191, 226, 247, 254, 255, 256, 264, 292, 348, 418, 474, 502, 510, 511, 512, 521, 557, 641, 767, 893, 977, 1013, 1022, 1023, 1024, 1034, 1079, 1199

Offset: 1

Juri-Stepan Gerasimov, Aug 06 2009

nonn

			a(1)=0!/(0!(0-0)!)=1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A008949.

Cf. A259525.

a163866 n = a163866_list !! (n-1)
a163866_list = scanl1 (+) $ concat a007318_tabl
-- Reinhard Zumkeller, Jul 18 2015

Flatten[Table[2^a-1+Sum[Binomial[a,p],{p,0,b}],{a,0,10},{b,0,a}]] (* Frank M Jackson, Apr 25 2011 *)
Accumulate[Flatten[Table[Binomial[n,k],{n,0,11},{k,0,n}]]] (* Harvey P. Dale, Dec 04 2012 *)

lista(nn) = {my(i=0, j=0, p=0); for (n=1, nn, p += binomial(i, j); print1(p, ", "); j++; if (j > i, j = 0; i++););} \\ Michel Marcus, Jan 25 2019

a(n) = Sum_{j=1..n-1} A007318(j).

Entries checked by R. J. Mathar, Aug 11 2009

A166454 Triangle read by rows: T(n, k) = (1/2)*(A007318(n,k) - A047999(n,k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 2, 5, 5, 2, 3, 7, 10, 7, 3, 3, 10, 17, 17, 10, 3, 4, 14, 28, 35, 28, 14, 4, 4, 18, 42, 63, 63, 42, 18, 4, 5, 22, 60, 105, 126, 105, 60, 22, 5, 5, 27, 82, 165, 231, 231, 165, 82, 27, 5, 6, 33, 110, 247, 396, 462, 396, 247, 110, 33, 6

Offset: 2

Gary W. Adamson, Oct 14 2009

Row sums = A120739: (1, 2, 7, 14, 30, 60, 127, 254, ...).

			First few rows of the triangle:
  1;
  1,   1;
  2,   3,   2;
  2,   5,   5,   2;
  3,   7,  10,   7,   3;
  3,  10,  17,  17,  10,   3;
  4,  14,  28,  35,  28,  14,   4;
  4,  18,  42,  63,  63,  42,  18,   4;
  5,  22,  60, 105, 126, 105,  60,  22,   5;
  5,  27,  82, 165, 231, 231, 165,  82,  27,   5;
  6,  33, 110, 247, 396, 462, 396, 247, 110,  33,   6;
  ...

Reinhard Zumkeller, Rows n = 2..125 of triangle, flattened

Cf. A047999, A120739.

Cf. A007318, A011848, A001700 (central terms).

Flat(List([2..12],n->List([1..n-1],m->Int(Binomial(n,m)/2)))); # Muniru A Asiru, Apr 14 2019

Following Bagula's formula
a166454 n k = a166454_tabl !! (n-2) !! (k-1)
a166454_row n = a166454_tabl !! (n-2)
a166454_tabl = map (map (flip div 2) . init . tail) $ drop 2 a007318_tabl
-- Reinhard Zumkeller, Mar 04 2015

[[Floor(Binomial(n,k)/2): k in [1..n-1]]: n in [2..12]]; // G. C. Greubel, Apr 16 2019

seq(seq(floor(binomial(n,m)/2),m=1..n-1),n=2..12); # Muniru A Asiru, Apr 14 2019

T[n_, m_] = Floor[Binomial[n, m]/2]; Table[T[n, m], {n, 2, 12}, {m, 1, n-1}]//Flatten (* Roger L. Bagula, Mar 07 2010*)

{T(n,k) = binomial(n,k)\2 };
for(n=2,12, for(k=1,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 16 2019

[[floor(binomial(n,k)/2) for k in (1..n-1)] for n in (2..12)] # G. C. Greubel, Apr 16 2019

T(n, k) = (1/2)*(A007318(n,k) - A047999(n,k)), nonzero terms.

T(n, m) = floor(binomial(n, m)/2). - Roger L. Bagula, Mar 07 2010

A256113 Table read by rows: T(1,1) = 1, for n > 1: row n = union of distinct prime factors occurring in terms of n-th row of Pascal's triangle, cf. A007318.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 5, 2, 3, 5, 3, 5, 7, 2, 5, 7, 2, 3, 7, 2, 3, 5, 7, 2, 3, 5, 7, 11, 2, 3, 5, 7, 11, 2, 3, 5, 11, 13, 2, 3, 7, 11, 13, 3, 5, 7, 11, 13, 2, 3, 5, 7, 11, 13, 2, 5, 7, 11, 13, 17, 2, 3, 5, 7, 11, 13, 17, 2, 3, 7, 11, 13, 17, 19, 2, 3, 5, 11, 13

Offset: 1

Reinhard Zumkeller, Mar 16 2015

			.  n |   T(n,k)   |                A001142(n) | A007318(n,0..n)
. ---+------------+---------------------------+-------------------------
.  1 | 1          |                         1 | 1  1
.  2 | 2          |                         2 | 1  2  1
.  3 | 3          |                         9 | 1  3  3   1
.  4 | 2 3        |                        96 | 1  4  6   4   1
.  5 | 2 5        |                      2500 | 1  5 10  10   5   1
.  6 | 2 3 5      |                    162000 | 1  6 15  20  15   6   1
.  7 | 3 5 7      |                  26471025 | 1  7 21  35  35  21   7   1
.  8 | 2 5 7      |               11014635520 | 1  8 28  56  70  56  28 ...
.  9 | 2 3 7      |            11759522374656 | 1  9 36  84 126 126  84 ...
. 10 | 2 3 5 7    |         32406091200000000 | 1 10 45 120 210 252 210 ...
. 11 | 2 3 5 7 11 |     231627686043080250000 | 1 11 55 165 330 462 462 ...
. 12 | 2 3 5 7 11 | 4311500661703860387840000 | 1 12 66 220 495 792 924 ...

Reinhard Zumkeller, b-file >>Rows n = 1..1000 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A027748, A001142, A004788 (row lengths), A056606 (row products).

a256113 n k = a256113_tabf !! (n-1) !! (n-1)
a256113_row n = a256113_tabf !! (n-1)
a256113_tabf = map a027748_row $ tail a001142_list

cp's OEIS Frontend

A132750 A132749 * [1, 2, 3, ...] = A007318 * A065190.

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A132823 A007318 + 2*A103451 - 2*A000012.

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A133084 A007318 * A133080.

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A133124 A007318 * [1, 2, 2, 3, 2, 3, 2, 3, 2, ...].

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A134083 A007318 * A134082.

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A135837 A007318 * a triangle with (1, 2, 2, 4, 4, 8, 8, ...) in the main diagonal and the rest zeros.

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Haskell

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A143690 a(n) = A007318 * [1, 6, 14, 9, 0, 0, 0, ...].

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A132823 A007318 + 2A103451 - 2A000012.