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A095666 Pascal (1,4) triangle. 16
4, 1, 4, 1, 5, 4, 1, 6, 9, 4, 1, 7, 15, 13, 4, 1, 8, 22, 28, 17, 4, 1, 9, 30, 50, 45, 21, 4, 1, 10, 39, 80, 95, 66, 25, 4, 1, 11, 49, 119, 175, 161, 91, 29, 4, 1, 12, 60, 168, 294, 336, 252, 120, 33, 4, 1, 13, 72, 228, 462, 630, 588, 372, 153, 37, 4, 1, 14, 85, 300, 690, 1092 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This is the fourth member, q=4, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1), A029635 (q=2) (but with a(0,0)=2, not 1), A095660.

This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column nr. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x) := Sum_{m=0..n} (a(n,m)*x^m is G(z,x) = g(z)/(1 - x*z*f(z)). Here: g(x) = (4-3*x)/(1-x), f(x) = 1/(1-x), hence G(z,x) = (4-3*z)/(1-(1+x)*z).

The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} a(n-1-k, k) = A022095(n-2), n >= 2, with n=1 value 4. [Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.]

T(2*n,n) = A029609(n) for n > 0, A029609 are the central terms of the Pascal (2,3) triangle A029600. [Reinhard Zumkeller, Apr 08 2012]

LINKS

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

W. Lang, First 10 rows.

FORMULA

Recursion: a(n, m) = 0 if m > n, a(0, 0) = 4; a(n, 0) = 1 if n>=1; a(n, m) = a(n-1, m) + a(n-1, m-1).

G.f. column m (without leading zeros): (4-3*x)/(1-x)^(m+1), m >= 0.

a(n,k) = (1 + 3*k/n)*binomial(n,k). [Mircea Merca, Apr 08 2012]

EXAMPLE

[4];

[1,4];

[1,5,4];

[1,6,9,4];

[1,7,15,13,4];

...

MAPLE

a(n, k):=(1+3*k/n)*binomial(n, k) # Mircea Merca, Apr 08 2012

PROG

(Haskell)

a095666 n k = a095666_tabl !! n !! k

a095666_row n = a095666_tabl !! n

a095666_tabl = [4] : iterate

   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1, 4]

-- Reinhard Zumkeller, Apr 08 2012

CROSSREFS

Row sums: A020714(n-1), n >= 1, 4 if n=0.

Alternating row sums are [4, -3, followed by 0's].

Column sequences (without leading zeros) give for m=1..9, with n >= 0: A000027(n+4), A055999(n+1), A060488(n+3), A095667-71, A095819.

Sequence in context: A126114 A323458 A074393 * A257231 A196757 A193254

Adjacent sequences:  A095663 A095664 A095665 * A095667 A095668 A095669

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jun 11 2004

STATUS

approved

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Last modified May 12 23:35 EDT 2021. Contains 343829 sequences. (Running on oeis4.)