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A100936 Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)*1, C(n,1)*2,..., C(n,k)*A051163(k), ..., C(n,n)*A051163(n)] and where A051162 equals the antidiagonal sums. 3
1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 76, 47, 11, 1, 1, 13, 71, 163, 163, 71, 13, 1, 1, 15, 100, 301, 435, 301, 100, 15, 1, 1, 17, 134, 502, 971, 971, 502, 134, 17, 1, 1, 19, 173, 778, 1909, 2577, 1909, 778, 173, 19, 1, 1, 21, 217, 1141 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Antidiagonal sums form A051163. Main diagonal is A100937. Different from A086620.

LINKS

Table of n, a(n) for n=0..69.

FORMULA

T(n, k) = Sum_{j=0..n} C(k, j)*C(n, j)*A051162(j), with T(0, 0) = 1 and where Sum_{i=0..n} T(n-i, i) = A051162(n).

EXAMPLE

Rows begin:

[1,1,1,1,1,1,1,1,1,...],

[1,3,5,7,9,11,13,15,17,...],

[1,5,14,28,47,71,100,134,...],

[1,7,28,76,163,301,502,778,...],

[1,9,47,163,435,971,1909,3417,...],

[1,11,71,301,971,2577,5917,12167,...],

[1,13,100,502,1909,5917,15678,36744,...],

[1,15,134,778,3417,12167,36744,97272,...],...

Antidiagonal sums form A051163: [1,2,5,12,30,76,194,496,1269,3250,8337,...].

The inverse binomial transform of the rows form the respective rows of the triangle B:

[1*1],

[1*1,1*2],

[1*1,2*2,1*5],

[1*1,3*2,3*5,1*12],

[1*1,4*2,6*5,4*12,1*30],...

where B(n,k) = binomial(n,k)*A051163(k).

PROG

(PARI) T(n, k)=if(n==0 || k==0, 1, sum(j=0, n, binomial(k, j)*binomial(n, j)*sum(i=0, j, T(j-i, i))); )

CROSSREFS

Cf. A051163, A100937.

Sequence in context: A144461 A106597 A108359 * A086620 A228356 A253670

Adjacent sequences:  A100933 A100934 A100935 * A100937 A100938 A100939

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 23 2004

STATUS

approved

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Last modified January 16 03:30 EST 2019. Contains 319184 sequences. (Running on oeis4.)