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A194949 Symmetric triangle T, read by rows, where the matrix product of T and T transpose yields a square array which, when read by antidiagonals, equals this triangle read by rows. 2
1, 1, 1, 2, 2, 2, 6, 4, 4, 6, 20, 10, 12, 10, 20, 72, 30, 28, 28, 30, 72, 260, 102, 84, 104, 84, 102, 260, 996, 362, 260, 268, 268, 260, 362, 996, 3772, 1358, 892, 832, 1144, 832, 892, 1358, 3772, 14852, 5130, 3236, 2928, 2956, 2956, 2928, 3236, 5130, 14852, 58204, 19982, 12044, 10072, 9948, 13736, 9948, 10072, 12044, 19982, 58204 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(n,k) = Sum_{j=0..k} T(n-k,j)*T(k,j) for n>0, k>=0, with T(0,0)=1.

Column 0 (A194950) equals row sums of triangle.

Central terms (A194951) equals sums of squares of terms in rows.

EXAMPLE

Triangle T begins:

1;

1, 1;

2, 2, 2;

6, 4, 4, 6;

20, 10, 12, 10, 20;

72, 30, 28, 28, 30, 72;

260, 102, 84, 104, 84, 102, 260;

996, 362, 260, 268, 268, 260, 362, 996;

3772, 1358, 892, 832, 1144, 832, 892, 1358, 3772;

14852, 5130, 3236, 2928, 2956, 2956, 2928, 3236, 5130, 14852;

58204, 19982, 12044, 10072, 9948, 13736, 9948, 10072, 12044, 19982, 58204; ...

...

Matrix product of T and T transpose, T*T~, yields the square array:

1, 1, 2, 6, 20, 72, 260, 996, 3772, ...;

1, 2, 4, 10, 30, 102, 362, 1358, 5130, ...;

2, 4, 12, 28, 84, 260, 892, 3236, 12044, ...;

6, 10, 28, 104, 268, 832, 2928, 10072, 36624, ...;

20, 30, 84, 268, 1144, 2956, 9948, 34700, 130924, ...;

72, 102, 260, 832, 2956, 13736, 36908, 124116, 454820, ...;

260, 362, 892, 2928, 9948, 36908, 180936, 488748, 1693572, ...;

996, 1358, 3236, 10072, 34700, 124116, 488748, 2524968, 6901788, ...;

3772, 5130, 12044, 36624, 130924, 454820, 1693572, 6901788, 36428808, ...;

...

which, when read by antidiagonals, equals this triangle read by rows.

PROG

(PARI) {T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, 1))); for(i=1, n, M=matrix(n+1, n+1, r, c, if(r>=c, if(c==1, if(r==1, 1, sum(j=1, r-1, (M*M~)[r-j, j])), (M*M~)[r-c+1, c])))); M[n+1, k+1]}

CROSSREFS

Cf. A194950, A194951.

Sequence in context: A209752 A119918 A084867 * A227550 A286384 A099259

Adjacent sequences:  A194946 A194947 A194948 * A194950 A194951 A194952

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 05 2011

STATUS

approved

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Last modified October 17 22:16 EDT 2019. Contains 328134 sequences. (Running on oeis4.)