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URL: https://oeis.org/A158824

⇱ A158824 - OEIS

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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
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.
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
(SageMath)
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
Row sums: A000332.
Sequence in context: A392271 A262370 A108759 * A348074 A334161 A039806
KEYWORD
nonn,tabl,easy
AUTHOR
STATUS
approved