A185630 -id:A185630 - cp's OEIS frontend

A185628 Triangle, read by rows, equal to the matrix cube of triangle A185620.

1, 3, 1, 6, 3, 1, 16, 12, 3, 1, 60, 55, 18, 3, 1, 305, 315, 118, 24, 3, 1, 1988, 2243, 912, 205, 30, 3, 1, 15951, 19378, 8342, 1995, 316, 36, 3, 1, 153171, 198363, 89193, 22274, 3708, 451, 42, 3, 1, 1722693, 2358860, 1098552, 283427, 48863, 6195, 610, 48, 3, 1, 22296396, 32060736

Offset: 0

Paul D. Hanna, Sep 07 2011

			 Triangle begins:
1;
3, 1;
6, 3, 1;
16, 12, 3, 1;
60, 55, 18, 3, 1;
305, 315, 118, 24, 3, 1;
1988, 2243, 912, 205, 30, 3, 1;
15951, 19378, 8342, 1995, 316, 36, 3, 1;
153171, 198363, 89193, 22274, 3708, 451, 42, 3, 1;
1722693, 2358860, 1098552, 283427, 48863, 6195, 610, 48, 3, 1;
22296396, 32060736, 15376126, 4070883, 720586, 94085, 9600, 793, 54, 3, 1; ...
This triangle equals the matrix cube, R^3, of triangle R = A185620, which begins:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 10, 5, 1, 1;
1, 42, 27, 7, 1, 1;
1, 226, 173, 52, 9, 1, 1;
1, 1525, 1330, 442, 85, 11, 1, 1;
1, 12555, 12134, 4345, 897, 126, 13, 1, 1; ...
where R^3 - R^2 + I equals R shifted left one column.

Cf. A185620, A185624, A185629, A185630, A185631.

A185629 Column 0 of triangle A185628; also, equals row sums of triangle A185624.

Original entry on oeis.org

1, 3, 6, 16, 60, 305, 1988, 15951, 153171, 1722693, 22296396, 327270440, 5381716953, 98134088419, 1967105567802, 43024511648982, 1020235144731236, 26083065249549660, 715430961891290714, 20962974995245684878, 653654392800091507791, 21615444263990028093227

Offset: 0

Paul D. Hanna, Sep 07 2011

nonn

Triangle A185628 equals the matrix cube of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Cf. A185620, A185628, A185630, A185631.

{a(n)=local(A=Mat(1), B); for(m=1, n+1, B=A^3-A^2+A^0; A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2|j==i, A[i, j]=1, if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return((A^3)[n+1, 1])}

A185631 Row sums of triangle A185628.

Original entry on oeis.org

1, 4, 10, 32, 137, 766, 5382, 46022, 467206, 5519252, 74629263, 1139283601, 19411643752, 365595475542, 7548810986515, 169681672611815, 4126880020198630, 108026180912533227, 3029184083924476085, 90617948406514693702, 2881333799993441893564, 97057552263114611503090

Offset: 0

Paul D. Hanna, Sep 07 2011

nonn

Triangle A185628 equals the matrix cube of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Cf. A185620, A185628, A185629, A185630.

{a(n)=local(A=Mat(1), B); for(m=1, n+1, B=A^3-A^2+A^0; A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2||j==i, A[i, j]=1, if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return(sum(k=1,n+1,(A^3)[n+1, k]))}

cp's OEIS Frontend

A185628 Triangle, read by rows, equal to the matrix cube of triangle A185620.

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A185629 Column 0 of triangle A185628; also, equals row sums of triangle A185624.

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A185631 Row sums of triangle A185628.

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