A003991 -id:A003991 - cp's OEIS frontend

A098360 Multiplication table of the cube numbers read by antidiagonals.

1, 8, 8, 27, 64, 27, 64, 216, 216, 64, 125, 512, 729, 512, 125, 216, 1000, 1728, 1728, 1000, 216, 343, 1728, 3375, 4096, 3375, 1728, 343, 512, 2744, 5832, 8000, 8000, 5832, 2744, 512, 729, 4096, 9261, 13824, 15625, 13824, 9261, 4096, 729, 1000, 5832, 13824

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

			1; 8,8; 27,64,27; 64,216,216,64; ...

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A003991, A098358, A098359, A098361.

Row sums: A145216. - N. J. A. Sloane, May 31 2009

Flat(List([2..11],m->List([1..m-1],i->i^3*(m-i)^3))); # Muniru A Asiru, Jun 27 2018

seq(seq(i^3*(m-i)^3,i=1..m-1),m=2..10); # Robert Israel, Jun 27 2018

With[{s = Range[10]^3}, Table[s[[#]] s[[j]] &[i - j + 1], {i, Length@s}, {j, i}]] // Flatten (* Michael De Vlieger, Jun 27 2018 *)

G.f. as rectangular array: [xy(1+4x+x^2)(1+4y+y^2)] / [(1-x)^4 * (1-y)^4 ]. - Ralf Stephan, Oct 27 2004, corrected by Robert Israel, Jun 27 2018

a(n) = A003991(n)^3.- Robert Israel, Jun 27 2018

More terms from Ralf Stephan, Oct 27 2004

Offset corrected by Robert Israel, Jun 27 2018

A319840 Table read by antidiagonals: T(n, k) is the number of elements on the perimeter of an n X k matrix.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 8, 8, 5, 6, 10, 10, 10, 10, 6, 7, 12, 12, 12, 12, 12, 7, 8, 14, 14, 14, 14, 14, 14, 8, 9, 16, 16, 16, 16, 16, 16, 16, 9, 10, 18, 18, 18, 18, 18, 18, 18, 18, 10, 11, 20, 20, 20, 20, 20, 20, 20, 20, 20, 11, 12, 22, 22, 22

Offset: 1

Stefano Spezia, Sep 29 2018

The table T(n, k) can be indifferently read by ascending or descending antidiagonals.

			The table T starts in row n=1 with columns k >= 1 as:
   1   2   3   4   5   6   7   8   9  10 ...
   2   4   6   8  10  12  14  16  18  20 ...
   3   6   8  10  12  14  16  18  20  22 ...
   4   8  10  12  14  16  18  20  22  24 ...
   5  10  12  14  16  18  20  22  24  26 ...
   6  12  14  16  18  20  22  24  26  28 ...
   7  14  16  18  20  22  24  26  28  30 ...
   8  16  18  20  22  24  26  28  30  32 ...
   9  18  20  22  24  26  28  30  32  34 ...
  10  20  22  24  26  28  30  32  34  36 ...
  ...
The triangle X(n, k) begins
  n\k|   1   2   3   4   5   6   7   8   9  10
  ---+----------------------------------------
   1 |   1
   2 |   2   2
   3 |   3   4   3
   4 |   4   6   6   4
   5 |   5   8   8   8   5
   6 |   6  10  10  10  10   6
   7 |   7  12  12  12  12  12   7
   8 |   8  14  14  14  14  14  14   8
   9 |   9  16  16  16  16  16  16  16   9
  10 |  10  18  18  18  18  18  18  18  18  10
  ...

Cf. A000027 (1st column/right diagonal of the triangle or 1st row/column of the table), A005843 (2nd row/column of the table, or 2nd column of the triangle), A008574 (main diagonal of the table), A005893 (row sum of the triangle).
Cf. A003991 (the number of elements in an n X k matrix).
Cf. A131821, A318274, A154325.

[[k lt 3 or n+1-k lt 3 select (n+1-k)*k else 2*n-2: k in [1..n]]: n in [1..10]]; // triangle output

a := (n, k) -> (n+1-k)*k-(n-1-k)*(k-2)*(limit(Heaviside(min(n+1-k, k)-3+x), x = 0, right)): seq(seq(a(n, k), k = 1 .. n), n = 1 .. 20)

Flatten[Table[(n + 1 - k) k-(n-1-k)*(k-2)Limit[HeavisideTheta[Min[n+1-k,k]-3+x], x->0, Direction->"FromAbove"  ],{n, 20}, {k, n}]] (* or *)
f[n_] := Table[SeriesCoefficient[(x y - x^3 y^3)/((-1 + x)^2 (-1 + y)^2), {x, 0, i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f,20]]

T(n, k) = if ((n+1-k<3) || (k<3), (n+1-k)*k, 2*n-2);
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(20) \\ triangle output

T(n, k) = n*k - (n - 2)*(k - 2)*H(min(n, k) - 3), where H(x) is the Heaviside step function, taking H(0) = 1.
G.f. as rectangular array: (x*y - x^3*y^3)/((-1 + x)^2*(-1 + y)^2).
X(n, k) = A131821(n, k)*A318274(n - 1, k)*A154325(n - 1, k). - Franck Maminirina Ramaharo, Nov 18 2018

A350066 Symmetric square array A(n,k) = A122111(A122111(n) * A122111(k)), n >= 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 7, 10, 10, 7, 6, 7, 10, 11, 9, 11, 10, 7, 8, 11, 14, 14, 14, 14, 11, 8, 9, 12, 13, 15, 13, 15, 13, 12, 9, 10, 15, 20, 22, 22, 22, 22, 20, 15, 10, 11, 14, 21, 18, 17, 21, 17, 18, 21, 14, 11, 12, 13, 22, 25, 28, 26, 26, 28, 25, 22, 13, 12, 13, 20, 17, 21, 33, 30, 19, 30, 33, 21, 17, 20, 13

Offset: 1

Antti Karttunen, Dec 13 2021

A122111 is a self-inverse permutation, so this array represents a binary operation A(.,.) over the positive integers that is isomorphic to multiplication. Its primes are the positive powers of 2 (as defined by standard multiplication): 2, 4, 8, 16, 32, ... . The positive powers of 2, as defined by A(.,.), are the prime numbers as we usually understand them: 2, 3, 5, 7, 11, ... . - Peter Munn, Aug 04 2022

			The top left 15 X 15 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  12, 13, 14,  15,
   2,  3,  5,  6,  7, 10, 11, 12, 15, 14, 13,  20, 17, 22,  21,
   3,  5,  7, 10, 11, 14, 13, 20, 21, 22, 17,  28, 19, 26,  33,
   4,  6, 10,  9, 14, 15, 22, 18, 25, 21, 26,  30, 34, 33,  35,
   5,  7, 11, 14, 13, 22, 17, 28, 33, 26, 19,  44, 23, 34,  39,
   6, 10, 14, 15, 22, 21, 26, 30, 35, 33, 34,  42, 38, 39,  55,
   7, 11, 13, 22, 17, 26, 19, 44, 39, 34, 23,  52, 29, 38,  51,
   8, 12, 20, 18, 28, 30, 44, 27, 50, 42, 52,  45, 68, 66,  70,
   9, 15, 21, 25, 33, 35, 39, 50, 49, 55, 51,  70, 57, 65,  77,
  10, 14, 22, 21, 26, 33, 34, 42, 55, 39, 38,  66, 46, 51,  65,
  11, 13, 17, 26, 19, 34, 23, 52, 51, 38, 29,  68, 31, 46,  57,
  12, 20, 28, 30, 44, 42, 52, 45, 70, 66, 68,  63, 76, 78, 110,
  13, 17, 19, 34, 23, 38, 29, 68, 57, 46, 31,  76, 37, 58,  69,
  14, 22, 26, 33, 34, 39, 38, 66, 65, 51, 46,  78, 58, 57,  85,
  15, 21, 33, 35, 39, 55, 51, 70, 77, 65, 57, 110, 69, 85,  91,

Index entries for sequences computed from indices in prime factorization

Cf. A122111, A297002 (main diagonal), A253550 (after its initial term, gives row 2 / column 2 from the second term onward).
See the formula section for the relationships with A003961, A061142.
Cf. also A003991, A129595, A331590.

up_to = 105;
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A350066sq(n,k) = A122111(A122111(n)*A122111(k));
A350066list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A350066sq(col,(a-(col-1))))); (v); };
v350066 = A350066list(up_to);
A350066(n) = v350066[n]; \\ Antti Karttunen, Dec 13 2021

A(n, A061142(n)) = A003961(n). - Peter Munn, Aug 04 2022

A081113 Number of paths of length n-1 a king can take from one side of an n X n chessboard to the opposite side.

Original entry on oeis.org

1, 4, 17, 68, 259, 950, 3387, 11814, 40503, 136946, 457795, 1515926, 4979777, 16246924, 52694573, 170028792, 546148863, 1747255194, 5569898331, 17698806798, 56076828573, 177208108824, 558658899825, 1757365514652

Offset: 1

David Scambler, Apr 16 2003

a(n) = number of sequences (a_1,a_2,...,a_n) with 1<=a_i<=n for all i and |a_(i+1)-a_(i)|<=1 for 1<=i<=n-1. For n=2 the sequences are 11, 12, 21, 22. - David Callan, Oct 24 2004

Simon Plouffe proposes the ordinary generating function A(x) (for offset zero) in the implicit form 3-10*x+12*x^2+(-4+30*x+54*x^3-72*x^2)*A(x)+(81*x^4+54*x^2+1-12*x-108*x^3)*A(x)^2 = 0 which delivers at least the first 200 terms (i.e., as far as tested) correctly. - David Scambler, R. J. Mathar, Jan 06 2011

			For n=2 the 4 paths are (0,0)->(0,1); (0,0)->(1,1); (1,0)->(0,1); (1,0)->(1,1).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Simon Plouffe, OEIS conjectured formulas.
D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I , arXiv:1612.08697 [math.CO], 2016-2017.

Cf. A005773 (paths which begin at a corner), diagonal of A296449.

A026300 := proc(n,k) add( binomial(n,2*i+n-k)*(binomial(2*i+n-k,i) -binomial(2*i+n-k,i-1)), i=0..floor(k/2)) ; end proc:
A081113 := proc(n) add(k*(n-k+1)*A026300(n-1,k-1),k=1..n) ; end proc:
seq(A081113(n),n=1..20) ;
# R. J. Mathar, Jun 09 2010

t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}] (* from A026300 *); f[n_] := Sum[ k(n - k + 1)t[n - 1, k - 1], {k, n}]; Array[f, 24]

a(n) = Sum_{k=1..n} k*(n-k+1)*M(n-1, k-1) where k*(n-k+1) is the triangular view of A003991 and M() is the Motzkin triangle A026300.
Conjecture: g.f.(x)=z*A064808(z), where z=x*A001006(x) and A...(x) are the corresponding generating functions. - R. J. Mathar, Jul 07 2009
Conjecture from WolframAlpha (verified for 1<=n<=180): (n+3)*a(n+4) = 27*n*a(n) +27*a(n+1) -9*(2*n+5)*a(n+2) +(8*n+21)*a(n+3). - Alexander R. Povolotsky, Jan 04 2011
Shorter recurrence: (n-1)*(2*n-7)*a(n) = (10*n^2-39*n+23)*a(n-1) - 3*(2*n^2-n-9)*a(n-2) - 9*(n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 28 2012
a(n) ~ 3^(n-1)*n*(1-4/(sqrt(3*Pi*n))). - Vaclav Kotesovec, Oct 28 2012
a(n) = (n+2)*3^(n-2)+2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corollary 2.8] - R. J. Mathar, Dec 13 2017

A098359 Multiplication table of the square numbers read by antidiagonals.

Original entry on oeis.org

1, 4, 4, 9, 16, 9, 16, 36, 36, 16, 25, 64, 81, 64, 25, 36, 100, 144, 144, 100, 36, 49, 144, 225, 256, 225, 144, 49, 64, 196, 324, 400, 400, 324, 196, 64, 81, 256, 441, 576, 625, 576, 441, 256, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 400, 729, 1024, 1225, 1296, 1225, 1024, 729, 400, 121

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k-1) = n. See A003991. - Charlie Marion, Apr 22 2013

			Square array A(n,k) begins:
   1,   4,   9,  16,   25,   36,   49, ...
   4,  16,  36,  64,  100,  144,  196, ...
   9,  36,  81, 144,  225,  324,  441, ...
  16,  64, 144, 256,  400,  576,  784, ...
  25, 100, 225, 400,  625,  900, 1225, ...
  36, 144, 324, 576,  900, 1296, 1764, ...
  49, 196, 441, 784, 1225, 1764, 2401, ...

Cf. A000290, A003991, A098358, A098360, A098361, A213547.
Antidiagonal sums give A033455.
Main diagonal gives A000583.

A:= (n,k)-> (n*k)^2:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, May 19 2025

A(n,k) = n^2*k^2.
G.f.: [xy(1+x)(1+y)] / [(1-x)^3 * (1-y)^3 ]. - Ralf Stephan, Oct 27 2004
Sum_{j=1..n} A(j,1+n-j)*j = A213547(n). - Alois P. Heinz, May 19 2025

Offset corrected by Alois P. Heinz, May 19 2025

A098364 Multiplication table of the digits of the square root of 2 read by antidiagonals.

Original entry on oeis.org

1, 4, 4, 1, 16, 1, 4, 4, 4, 4, 2, 16, 1, 16, 2, 1, 8, 4, 4, 8, 1, 3, 4, 2, 16, 2, 4, 3, 5, 12, 1, 8, 8, 1, 12, 5, 6, 20, 3, 4, 4, 4, 3, 20, 6, 2, 24, 5, 12, 2, 2, 12, 5, 24, 2, 3, 8, 6, 20, 6, 1, 6, 20, 6, 8, 3, 7, 12, 2, 24, 10, 3, 3, 10, 24, 2, 12, 7, 3, 28, 3, 8, 12, 5, 9, 5, 12, 8, 3, 28, 3

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

			Triangle begins:
  1;
  4,4;
  1,16,1;
  4,4,4,4;
  ...
Array begins:
  1  4 1  4 2 ...
  4 16 4 16 8 ...
  1  4 1  4 2 ...
  4 16 4 16 8 ...
  2  8 2  8 4 ...
  ...

Michel Marcus, Antidiagonals n = 1..100, flattened

Cf. A002193, A003991, A098365, A098366, A098367.

sqrt2(nn) = {my(r=0, x=2, list = List(), d); for(digits=1, nn, d=0; while((20*r+d)*d <= x, d++); d--; listput(list, d); x=100*(x-(20*r+d)*d); r=10*r+d;); Vec(list);} \\ A002193
lista(nn) = {my(dd = sqrt2(nn)); for (n=1, nn, for (k=1, n, print1(dd[k]*dd[n-k+1], ", ")));} \\ Michel Marcus, Nov 11 2021

T(n,k) = A003991(A002193(n), A002193(k)). - Michel Marcus, Nov 03 2021

Offset changed to 1 and a(34)=1 inserted by Georg Fischer, Nov 02 2021

A158823 Triangle read by rows: matrix product A004736 * A158821.

Original entry on oeis.org

1, 3, 1, 6, 2, 2, 10, 3, 4, 3, 15, 4, 6, 6, 4, 21, 5, 8, 9, 8, 5, 28, 6, 10, 12, 12, 10, 6, 36, 7, 12, 15, 16, 15, 12, 7, 45, 8, 14, 18, 20, 20, 18, 14, 8, 55, 9, 16, 21, 24, 25, 24, 21, 16, 9, 66, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 78, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11

Offset: 1

Gary W. Adamson & Roger L. Bagula, Mar 28 2009

			First few rows of the triangle =
   1;
   3,  1;
   6,  2,  2;
  10,  3,  4,  3;
  15,  4,  6,  6,  4;
  21,  5,  8,  9,  8,  5;
  28,  6, 10, 12, 12, 10,  6;
  36,  7, 12, 15, 16, 15, 12,  7;
  45,  8, 14, 18, 20, 20, 18, 14,  8;
  55,  9, 16, 21, 24, 25, 24, 21, 16,  9;
  66, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10;
  78, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11;
  91, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000292 (row sums), A003991, A004736, A158821.

[k eq 1 select Binomial(n+1, 2) else (n-k+1)*(k-1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 01 2021

A158823 := proc(n,m) add( A004736(n,k)*A158821(k-1,m-1),k=1..n) ; end: seq(seq(A158823(n,m),m=1..n),n=1..8) ; # R. J. Mathar, Oct 22 2009

Table[If[k==1, Binomial[n+1, 2], (n-k+1)*(k-1)], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

flatten([[binomial(n+1, 2) if k==1 else (n-k+1)*(k-1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 01 2021

Sum_{k=1..n} T(n, k) = A000292(n).
T(n, k) = Sum_{j=k..n} A004736(n, j)*A158821(j-1, k-1).
From R. J. Mathar, Mar 03 2011: (Start)
T(n, k) = (n-k+1)*(k-1), k>1.
T(n, 1) = A000217(n). (End)

Corrected A-number in a formula - R. J. Mathar, Oct 30 2009

A173997 Irregular triangle by columns derived from (1, 2, 3, ...) * (1, 2, 3, ...).

Original entry on oeis.org

1, 2, 3, 2, 4, 4, 5, 6, 3, 6, 8, 6, 7, 10, 9, 4, 8, 12, 12, 8, 9, 14, 15, 12, 5, 10, 16, 18, 16, 10, 11, 18, 21, 20, 15, 6, 12, 20, 24, 24, 20, 12, 13, 22, 27, 28, 25, 18, 7, 14, 24, 30, 32, 30, 24, 14, 15, 26, 33, 36, 35, 30, 21, 8, 16, 28, 36, 40, 40, 36, 28, 16

Offset: 1

Gary W. Adamson, Mar 05 2010

Given a (1, 2, 3, ...) * (1, 2, 3, ...) multiplication table; leftmost column of the triangle = (1, 2, 3, ...). Then shift down each successive column of the array twice to get this irregular triangle.

			Given:
  1,  2,  3,  4,  5, ...
  2,  4,  6,  8, 10, ...
  3,  6,  9, 12, 15, ...
  4,  8, 12, 16, 20, ...
  ...
After the shift twice operation, we obtain:
   1;
   2;
   3,  2;
   4,  4;
   5,  6,  3;
   6,  8,  6;
   7, 10,  9,  4;
   8, 12, 12,  8;
   9, 14, 15, 12,  5;
  10, 16, 18, 16, 10;
  11, 18, 21, 20, 15,  6;
  12, 20, 24, 24, 20, 12;
  ...

Stefano Spezia, First 200 rows of the triangle, flattened

Cf. A003991, A006918 (row sums).

Flatten[Table[k(2-2k+n),{n,16},{k,Floor[(n+1)/2]}]] (* Stefano Spezia, Apr 19 2022 *)

T(n, k) = k*(2 - 2*k + n), with 1 <= k <= floor((n + 1)/2). - Stefano Spezia, Apr 19 2022

A185904 Multiplication table for the tetrahedral numbers (A000292), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 10, 16, 10, 20, 40, 40, 20, 35, 80, 100, 80, 35, 56, 140, 200, 200, 140, 56, 84, 224, 350, 400, 350, 224, 84, 120, 336, 560, 700, 700, 560, 336, 120, 165, 480, 840, 1120, 1225, 1120, 840, 480, 165, 220, 660, 1200, 1680, 1960, 1960, 1680, 1200, 660, 220, 286, 880, 1650, 2400, 2940, 3136, 2940, 2400, 1650, 880, 286, 364, 1144, 2200, 3300, 4200, 4704, 4704, 4200, 3300, 2200, 1144, 364, 455, 1456, 2860, 4400, 5775, 6720, 7056, 6720, 5775, 4400, 2860, 1456, 455, 560, 1820, 3640, 5720, 7700, 9240, 10080, 10080, 9240, 7700, 5720, 3640, 1820, 560

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185906 < A000007 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,  4,  10,  20,  35
   4, 16,  40,  80, 140
  10, 40, 100, 200, 350
  20, 80, 200, 400, 700

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

Cf. A000007, A003991, A098358, A144112, A185905, A185906, A185907.

Row 1 = Column 1 = A000292.

(* This program generates A098358 and its accumulation array, A185904. *)
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A098358 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185904 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185904 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
T[n_, k_] := Binomial[k + 2, 3]*Binomial[n + 2, 3]; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

T(n,k) = binomial(k+2,3)*binomial(n+2,3), k >= 1, n >= 1.

A185905 Rectangular array binomial(k+3,4)*binomial(n+3,4), by antidiagonals.

Original entry on oeis.org

1, 5, 5, 15, 25, 15, 35, 75, 75, 35, 70, 175, 225, 175, 70, 126, 350, 525, 525, 350, 126, 210, 630, 1050, 1225, 1050, 630, 210, 330, 1050, 1890, 2450, 2450, 1890, 1050, 330, 495, 1650, 3150, 4410, 4900, 4410, 3150, 1650, 495, 715, 2475, 4950, 7350, 8820, 8820, 7350, 4950, 2475, 715, 1001, 3575, 7425, 11550

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185906 < A000007 < A000012 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,    5,   15,   35,   70
   5,   25,   75,  175,  350
  15,   75,  225,  525, 1050
  35,  175,  425, 1225, 2450

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

Cf. A144112.

Row 1 = Column 1 = A000332.

a[n_, k_] := Binomial[k + 3, 4]*Binomial[n + 3, 4]; Table[a[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

T(n,k) = binomial(k+3,4)*binomial(n+3,4), k >= 1, n >= 1.

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