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A158824 Triangle T(n,k) = A000292(n) if k = 1 otherwise (k-1)*(n-k+1)*(n-k+2)/2, read by rows. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A334161 A039806 A030320

Adjacent sequences:  A158821 A158822 A158823 * A158825 A158826 A158827

KEYWORD

nonn,tabl,easy

AUTHOR

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

STATUS

approved

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Last modified April 20 01:20 EDT 2021. Contains 343117 sequences. (Running on oeis4.)