A171631 Triangle read by rows: T(n,k) = n*(binomial(n-2, k-1) + n*binomial(n-2, k)), n > 0 and 0 <= k <= n - 1.

1, 4, 2, 9, 12, 3, 16, 36, 24, 4, 25, 80, 90, 40, 5, 36, 150, 240, 180, 60, 6, 49, 252, 525, 560, 315, 84, 7, 64, 392, 1008, 1400, 1120, 504, 112, 8, 81, 576, 1764, 3024, 3150, 2016, 756, 144, 9, 100, 810, 2880, 5880, 7560, 6300, 3360, 1080, 180, 10, 121, 1100

Offset

1,2

Comments

If T(0,0) = 0 is prepended, then row sums give A001788.

References

Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, 1945, p. 32.

Links

Table of n, a(n) for n=1..57.

Formula

Let p(x;n) = (x + 1)^n. Then row n are the coefficients in the expansion of p''(x;n) - x*p'(x;n) + n*p(x;n) = n*(x + n)*(x + 1)^(n - 2).

From Franck Maminirina Ramaharo, Oct 02 2018: (Start)

T(n,1) = A000290(n).

T(n,2) = A011379(n).

T(n,3) = 3*A002417(n-2).

T(n,n-2) = A046092(n-1).

T(n,n-3) = 9*A000292(n-2).

G.f.: y*(x*y - y - 1)/(x*y + y - 1)^3. (End)

Example

Triangle begins:
n\k|  0    1     2     3     4     6    7    8  9
-------------------------------------------------
1  |  1
2  |  4    2
3  |  9   12     3
4  | 16   36    24     4
5  | 25   80    90    40     5
6  | 36  150   240   180    60     6
7  | 49  252   525   560   315    84    7
8  | 64  392  1008  1400  1120   504  112    8
9  | 81  576  1764  3024  3150  2016  756  144  9
... reformatted. - Franck Maminirina Ramaharo, Oct 02 2018

Mathematica

Table[CoefficientList[n*(x + n)*(x + 1)^(n - 2), x], {n, 1, 12}]//Flatten

Prog

(Maxima) T(n, k) := n*(binomial(n - 2, k - 1) + n*binomial(n - 2, k))$
tabl(nn) := for n:1 thru nn do print(makelist(T(n, k), k, 0, n - 1))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */

Crossrefs

Cf. A003506, A007318, A127952, A171531.

Sequence in context: A233295 A298567 A006172 * A052915 A130273 A016516

Adjacent sequences: A171628 A171629 A171630 * A171632 A171633 A171634

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Dec 13 2009

Extensions

Edited and new name by Franck Maminirina Ramaharo, Oct 02 2018

Status

approved