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A096940 Pascal (1,5) triangle. 13
5, 1, 5, 1, 6, 5, 1, 7, 11, 5, 1, 8, 18, 16, 5, 1, 9, 26, 34, 21, 5, 1, 10, 35, 60, 55, 26, 5, 1, 11, 45, 95, 115, 81, 31, 5, 1, 12, 56, 140, 210, 196, 112, 36, 5, 1, 13, 68, 196, 350, 406, 308, 148, 41, 5, 1, 14, 81, 264, 546, 756, 714, 456, 189, 46, 5, 1, 15, 95, 345, 810, 1302 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This is the fifth member, q=5, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1), A029635 (q=2) (but with a(0,0)=2, not 1), A095660, A095666.

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)=(5-4*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(5-4*z)/(1-(1+x)*z).

The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} a(n-1-k, k) = A022096(n-2), n>=2, with n=1 value 5. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

LINKS

David A. Corneth, Table of n, a(n) for n = 0..9999

W. Lang, First 10 rows.

FORMULA

Recursion: a(n, m)=0 if m>n, a(0, 0)= 5; 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): (5-4*x)/(1-x)^(m+1), m>=0.

a(n,k) = (1+4*k/n)*binomial(n,k), for n>0. - Mircea Merca, Apr 08 2012

EXAMPLE

Triangle begins:

  5;

  1,  5;

  1,  6,  5;

  1,  7, 11,   5;

  1,  8, 18,  16,   5;

  1,  9, 26,  34,  21,   5;

  1, 10, 35,  60,  55,  26,   5;

  1, 11, 45,  95, 115,  81,  31,   5;

  1, 12, 56, 140, 210, 196, 112,  36,   5;

  1, 13, 68, 196, 350, 406, 308, 148,  41,  5;

  1, 14, 81, 264, 546, 756, 714, 456, 189, 46, 5; etc.

MAPLE

a(n, k):=piecewise(n=0, 5, 0<n, (1+4*k/n)*binomial(n, k)) # Mircea Merca, Apr 08 2012

PROG

(PARI) a(n) = {if(n <= 1, return(5 - 4*(n==1))); my(m = (sqrtint(8*n + 1) - 1)\2, t = n - binomial(m + 1, 2)); (1+4*t/m)*binomial(m, t)} \\ David A. Corneth, Aug 28 2019

CROSSREFS

Row sums: A007283(n-1), n>=1, 5 if n=0; g.f.: (5-4*x)/(1-2*x). Alternating row sums are [5, -4, followed by 0's].

Column sequences (without leading zeros) give for m=1..9, with n>=0: A000027(n+5), A056000(n-1), A096941-7.

Sequence in context: A095118 A251417 A100947 * A141345 A318664 A329031

Adjacent sequences:  A096937 A096938 A096939 * A096941 A096942 A096943

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jul 16 2004

STATUS

approved

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Last modified February 19 19:06 EST 2020. Contains 332047 sequences. (Running on oeis4.)