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A081581 Pascal-(1,6,1) array. 11
1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 22, 78, 22, 1, 1, 29, 190, 190, 29, 1, 1, 36, 351, 848, 351, 36, 1, 1, 43, 561, 2339, 2339, 561, 43, 1, 1, 50, 820, 5006, 9766, 5006, 820, 50, 1, 1, 57, 1128, 9192, 28806, 28806, 9192, 1128, 57, 1, 1, 64, 1485, 15240, 68034, 116208 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016993, A081591, A081592. Coefficients of the row polynomials in the Newton basis are given by A013614.

LINKS

Vincenzo Librandi, Rows n = 0..100, flattened

FORMULA

Square array T(n, k) defined by T(n, 0)=T(0, k)=1, T(n, k)=T(n, k-1)+6T(n-1, k-1)+T(n-1, k). Rows are the expansions of (1+6x)^k/(1-x)^(k+1).

E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(7*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 14*x + 49*x^2/2) = 1 + 15*x + 78*x^2/2! + 190*x^3/3! + 351*x^4/4! + 561*x^5/5! + .... - Peter Bala, Mar 05 2017

EXAMPLE

Rows start

1 1 1 1 1 ....

1 8 15 22 29 ....

1 15 78 190 351 ...

1 22 190 848 2339 ...

1 29 351 2339 9766 ...

As triangle, starts

1,

1, 1,

1, 8, 1,

1, 15, 15, 1,

1, 22, 78, 22, 1,

1, 29, 190, 190, 29, 1,

1, 36, 351, 848, 351, 36, 1,

1, 43, 561, 2339, 2339, 561, 43, 1,

...

MATHEMATICA

Table[ Hypergeometric2F1[-k, k-n, 1, 7], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, May 24 2013 *)

PROG

(PARI) t(n, k) = sum(j=0, n-k, binomial(n-k, j)*binomial(k, j)*7^j) \\ Michel Marcus, May 24 2013

CROSSREFS

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081582 (m = 7), A143683 (m = 8).

Sequence in context: A131067 A157170 A143679 * A174125 A051425 A051469

Adjacent sequences:  A081578 A081579 A081580 * A081582 A081583 A081584

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Mar 23 2003

STATUS

approved

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Last modified July 10 00:25 EDT 2020. Contains 335570 sequences. (Running on oeis4.)