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A081582 Pascal-(1,7,1) array. 12
1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 97, 25, 1, 1, 33, 241, 241, 33, 1, 1, 41, 449, 1161, 449, 41, 1, 1, 49, 721, 3297, 3297, 721, 49, 1, 1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1, 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1, 1, 73, 1921, 22121, 108353 (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 A017077, A081593, A081594. Coefficients of the row polynomials in the Newton basis are given by A013614.

LINKS

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

FORMULA

T(n,k) = Sum_{j = 0..n-k) binomial(n-k,j)*binomial*(k,j)*8^j. Riordan array (1/(1 - x), x*(1 + 7*x)/(1 - x)).

Square array T(n, k) defined by T(n, 0) = T(0, k)=1, T(n, k) = T(n, k-1) + 7*T(n-1, k-1) + T(n-1, k).

Rows are the expansions of (1 + 7*x)^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)*(8*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 16*x + 64*x^2/2) = 1 + 17*x + 97*x^2/2! + 241*x^3/3! + 449*x^4/4! + 721*x^5/5! + .... - Peter Bala, Mar 05 2017

EXAMPLE

Rows begin

1 1 1 1 1 ....

1 9 17 25 33 ....

1 17 97 241 449 ...

1 25 241 1161 3297 ...

1 33 449 3297 14721 ...

Triangle begins

1,

1, 1,

1, 9, 1,

1, 17, 17, 1,

1, 25, 97, 25, 1,

1, 33, 241, 241, 33, 1,

1, 41, 449, 1161, 449, 41, 1,

1, 49, 721, 3297, 3297, 721, 49, 1,

1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1,

...

MATHEMATICA

Table[ Hypergeometric2F1[-k, k-n, 1, 8], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, 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), A081581 (m = 6), A143683 (m = 8).

Sequence in context: A128060 A168625 A143681 * A174346 A144404 A014761

Adjacent sequences:  A081579 A081580 A081581 * A081583 A081584 A081585

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Mar 23 2003

STATUS

approved

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Last modified August 26 05:23 EDT 2019. Contains 326328 sequences. (Running on oeis4.)