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A109244 A tree-node counting triangle. 3
1, 1, 1, 4, 2, 1, 13, 7, 3, 1, 46, 24, 11, 4, 1, 166, 86, 40, 16, 5, 1, 610, 314, 148, 62, 22, 6, 1, 2269, 1163, 553, 239, 91, 29, 7, 1, 8518, 4352, 2083, 920, 367, 128, 37, 8, 1, 32206, 16414, 7896, 3544, 1461, 541, 174, 46, 9, 1, 122464, 62292, 30086, 13672, 5776, 2232 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Columns include A026641,A014300,A014301. Inverse matrix is A109246. Row sums are A014300. Diagonal sums are A109245.

LINKS

G. C. Greubel, Rows n=0..100 of triangle, flattened

FORMULA

Number triangle T(n, k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n+i-k, i-k).

Riordan array (1/(1-x*c(x)-2*x^2*c(x)^2), x*c(x)) where c(x)=g.f. of A000108.

The production matrix M (discarding the zeros) is:

1, 1;

3, 1, 1;

3, 1, 1, 1;

3, 1, 1, 1, 1;

... such that the n-th row of the triangle is the top row of M^n. - Gary W. Adamson, Feb 16 2012

EXAMPLE

Rows begin

1;

1,1;

4,2,1;

13,7,3,1;

46,24,11,4,1;

166,86,40,16,5,1;

MATHEMATICA

Table[Sum[(-1)^(n-j)*Binomial[n+j-k, j-k], {j, 0, n}], {n, 0, 12}, {k, 0, n}] //Flatten  (* G. C. Greubel, Feb 19 2019 *)

PROG

(PARI) {T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n+j-k, j-k))};

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 19 2019

(MAGMA) [[(&+[(-1)^(n-j)*Binomial(n+j-k, j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 19 2019

(Sage) [[sum((-1)^(n-j)*binomial(n+j-k, j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 19 2019

(GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(n-j)*Binomial(n+j-k, j-k) )))); # G. C. Greubel, Feb 19 2019

CROSSREFS

Sequence in context: A152818 A302235 A242861 * A171650 A225476 A143777

Adjacent sequences:  A109241 A109242 A109243 * A109245 A109246 A109247

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jun 23 2005

STATUS

approved

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Last modified June 25 03:50 EDT 2019. Contains 324338 sequences. (Running on oeis4.)