A166908 -id:A166908

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A166905

Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

+10
4

1, 1, 1, 6, 4, 1, 54, 33, 9, 1, 640, 380, 108, 16, 1, 9380, 5510, 1610, 270, 25, 1, 163576, 95732, 28560, 5148, 570, 36, 1, 3305484, 1933288, 586320, 110929, 13650, 1071, 49, 1, 75915708, 44437080, 13658904, 2677008, 353600, 31624, 1848, 64, 1, 1952409954

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

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..45.

EXAMPLE

Triangle begins:

1,1;

6,4,1;

54,33,9,1;

640,380,108,16,1;

9380,5510,1610,270,25,1;

163576,95732,28560,5148,570,36,1;

3305484,1933288,586320,110929,13650,1071,49,1;

75915708,44437080,13658904,2677008,353600,31624,1848,64,1;

1952409954,1144564278,355787568,71648322,9962949,973845,66150,2988,81,1;

55573310936,32638644236,10243342296,2107966432,304857190,31795560,2395120,127720,4590,100,1;

...

Coefficients in iterations of x*Catalan(x) form table A158825:

1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;

1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;

1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;

1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;

1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;

1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;

1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;

...

This triangle T transforms rows into diagonals of A158825;

the initial diagonals begin:

A158831: [1,1,6,54,640,9380,163576,3305484,...];

A158832: [1,2,12,110,1330,19852,351792,7209036,...];

A158833: [1,3,20,195,2464,38052,693048,14528217,...];

A158834: [1,4,30,315,4200,67620,1273668,27454218,...].

For example:

T * [1,0,0,0,0,0,0,0,0,0,0,0,0, ...] = A158831;

T * [1,1,2,5,14,42,132,429,1430,...] = A158832;

T * [1,2,6,21,80,322,1348,5814, ...] = A158833;

T * [1,3,12,54,260, 1310, 6824, ...] = A158834.

PROG

(PARI) {T(n, k)=local(F=x, G=serreverse(x-x^2+O(x^(n+3))), M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}

CROSSREFS

Cf. A166906, A166907, A166908, A166909, variant: A166900.

Cf. A158825, A158831, A158832, A158833, A158834, A158835, A000108.

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 28 2009

STATUS

approved

A166906

Column 1 of triangle A166905.

+10
4

1, 4, 33, 380, 5510, 95732, 1933288, 44437080, 1144564278, 32638644236, 1020503373032, 34708182795156, 1275532011982176, 50365443858930384, 2126358227959866224, 95577781657788563192, 4556923094838105968302

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

OFFSET

0,2

COMMENTS

Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

LINKS

Table of n, a(n) for n=0..16.

PROG

(PARI) {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+4))), M, N, P, m=n); M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); N=matrix(m+2, m+2, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); P=matrix(m+2, m+2, r, c, M[r+1, c]); (P~*N~^-1)[n+2, 2]}

CROSSREFS

Cf. A166905, A158831, A166907, A166908, A166909.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 28 2009

STATUS

approved

A166907

Column 2 of triangle A166905.

+10
4

1, 9, 108, 1610, 28560, 586320, 13658904, 355787568, 10243342296, 322939137312, 11063339361360, 409194048521778, 16249995494795920, 689585033717023224, 31140529927119263136, 1490994828293677370148, 75444108490820383882392

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

OFFSET

0,2

COMMENTS

Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

LINKS

Table of n, a(n) for n=0..16.

PROG

(PARI) {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+5))), M, N, P, m=n); M=matrix(m+4, m+4, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+4)))); polcoeff(F, c)); N=matrix(m+3, m+3, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+4)))); polcoeff(F, c)); P=matrix(m+3, m+3, r, c, M[r+1, c]); (P~*N~^-1)[n+3, 3]}

CROSSREFS

Cf. A166905, A158831, A166906, A166908, A166909.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 28 2009

STATUS

approved

A166909

Row sums of triangle A166905.

+10
4

1, 2, 11, 97, 1145, 16796, 293623, 5950792, 137075837, 3535416136, 100902444181, 3156570232069, 107392381479683, 3947409366073512, 155880018189733841, 6581149438442041483, 295807451972657856921, 14102499966460374953016

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

OFFSET

0,2

COMMENTS

Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

LINKS

Table of n, a(n) for n=0..17.

PROG

(PARI) {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+3))), M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x; for(i=1, r, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); P=matrix(n+1, n+1, r, c, M[r+1, c]); M=(P~*N~^-1); sum(k=1, n+1, M[n+1, k])}

CROSSREFS

Cf. A166905, A158831, A166906, A166907, A166908.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 28 2009

STATUS

approved

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