A247236 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A247236

Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+k)^k.

1, -1, 2, -1, -10, 3, -1, 26, -33, 4, -1, -54, 207, -76, 5, -1, 96, -993, 824, -145, 6, -1, -156, 4047, -6736, 2375, -246, 7, -1, 236, -14769, 46184, -28985, 5634, -385, 8, -1, -340, 49743, -280408, 293575, -95166, 11711, -568, 9, -1, 470, -157617, 1556672, -2609465, 1322334, -260449, 22112, -801, 10

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^0 + T(n,1)*(x+1)^1 + T(n,2)*(x+2)^2 + ... + T(n,n)*(x+n)^n, for n >= 0.

LINKS

Table of n, a(n) for n=0..54.

FORMULA

T(n,n) = n+1 = A000027(n+1), n >= 0.

T(n,1) = ((-1)^n*(1-4*n^3-10*n^2-4*n)-1)/8 = 2*(-1)^(n+1)*A002717(n), for n >= 1.

T(n,n-1) = n - n^2 - n^3 (A085490), for n >= 1.

T(n,n-2) = (n^5-2*n^4-3*n^3+6*n^2-2)/2, for n >= 2.

EXAMPLE

From Wolfdieter Lang, Jan 12 2015: (Start)

The triangle T(n,k) starts:

n\k 0 1 2 3 4 5 6 7 8 9 ...

0: 1

1: -1 2

2: -1 -10 3

3: -1 26 -33 4

4: -1 -54 207 -76 5

5: -1 96 -993 824 -145 6

6: -1 -156 4047 -6736 2375 -246 7

7: -1 236 -14769 46184 -28985 5634 -385 8

8: -1 -340 49743 -280408 293575 -95166 11711 -568 9

9: -1 470 -157617 1556672 -2609465 1322334 -260449 22112 -801 10

... Reformatted.

---------------------------------------------------------------------

n=3: 1 + 2*x + 3*x^2 + 4*x^3 = -1*(x+0)^0 + 26*(x+1)^1 - 33*(x+2)^2 + 4*(x+3)^3. (End)

PROG

(PARI) T(n, k)=(k+1)-sum(i=k+1, n, i^(i-k)*binomial(i, k)*T(n, i))

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

CROSSREFS

Cf. A248345, A000027, A002717, A085490.

Sequence in context: A110169 A144274 A144275 * A011268 A332080 A225911

Adjacent sequences: A247233 A247234 A247235 * A247237 A247238 A247239

KEYWORD

sign,tabl

AUTHOR

Derek Orr, Nov 27 2014

EXTENSIONS

Edited. - Wolfdieter Lang, Jan 12 2015

STATUS

approved