A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)*(n-k+1)*(n-k+2)/2, read by rows.

1, 4, 1, 10, 3, 2, 20, 6, 6, 3, 35, 10, 12, 9, 4, 56, 15, 20, 18, 12, 5, 84, 21, 30, 30, 24, 15, 6, 120, 28, 42, 45, 40, 30, 18, 7, 165, 36, 56, 63, 60, 50, 36, 21, 8, 220, 45, 72, 84, 84, 75, 60, 42, 24, 9, 286, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10, 364, 66, 110, 135, 144, 140, 126, 105, 80, 54, 30, 11

Offset

1,2

Comments

The triangle can also be defined by multiplying the triangles A(n,k)=1 and A158823(n,k), that is, this here are the partial column sums of A158823.

Links

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

Formula

T(n,k) = binomial(n+2,3) if k = 1 otherwise (k-1)*binomial(n-k+2, 2).

Sum_{k=1..n} T(n, k) = binomial(n+3, 4) = A000332(n+3). - G. C. Greubel, Apr 01 2021

Example

First few rows of the triangle are:
    1;
    4,  1;
   10,  3,   2;
   20,  6,   6,   3;
   35, 10,  12,   9,   4;
   56, 15,  20,  18,  12,   5;
   84, 21,  30,  30,  24,  15,   6;
  120, 28,  42,  45,  40,  30,  18,   7;
  165, 36,  56,  63,  60,  50,  36,  21,   8;
  220, 45,  72,  84,  84,  75,  60,  42,  24,  9;
  286, 55,  90, 108, 112, 105,  90,  70,  48, 27, 10;
  364, 66, 110, 135, 144, 140, 126, 105,  80, 54, 30, 11;
  455, 78, 132, 165, 180, 180, 168, 147, 120, 90, 60, 33, 12;
  ...

Mathematica

T[n_, k_]:= If[k==1, Binomial[n+2, 3], (k-1)*Binomial[n-k+2, 2]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

Prog

(Magma) A158824:= func< n, k | k eq 1 select Binomial(n+2, 3) else (k-1)*Binomial(n-k+2, 2) >; [A158824(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021
(Sage)
def A158824(n, k): return binomial(n+2, 3) if k==1 else (k-1)*binomial(n-k+2, 2)
flatten([[A158824(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

Crossrefs

Cf. A062707, A104633, A158823.

Row sums: A000332.

Sequence in context: A006370 A262370 A108759 * A348074 A334161 A039806

Adjacent sequences: A158821 A158822 A158823 * A158825 A158826 A158827

Keyword

nonn,tabl,easy

Author

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

Status

approved