A095663 -id:A095663 - cp's OEIS frontend

A095660 Pascal (1,3) triangle.

3, 1, 3, 1, 4, 3, 1, 5, 7, 3, 1, 6, 12, 10, 3, 1, 7, 18, 22, 13, 3, 1, 8, 25, 40, 35, 16, 3, 1, 9, 33, 65, 75, 51, 19, 3, 1, 10, 42, 98, 140, 126, 70, 22, 3, 1, 11, 52, 140, 238, 266, 196, 92, 25, 3, 1, 12, 63, 192, 378, 504, 462, 288, 117, 28, 3, 1, 13, 75, 255, 570, 882, 966, 750, 405, 145, 31, 3

Offset: 0

Wolfdieter Lang, May 21 2004

This is the third member, q=3, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1)), A029635 (q=2) (but with T(0,0)=2, not 1).
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 no. 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} T(n,m)*x^m is G(z,x) = g(z)/(1-x*z*f(z)). Here: g(x) = (3-2*x)/(1-x), f(x) = 1/(1-x), hence G(z,x) = (3-2*z)/(1-(1+x)*z).
The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} T(n-1-k,k) = A000285(n-2), n>=2, with n=1 value 3. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Central terms: T(2*n,n) = A028329(n) = A100320(n) for n > 0, A028329 are the central terms of triangle A028326. - Reinhard Zumkeller, Apr 08 2012
Let P be Pascal's triangle, A007318 and R the Riordan array, A097805. Then Pascal triangle (1,q) = ((q-1) * R) + P. Example: Pascal triangle (1,3) = (2 * R) + P. - Gary W. Adamson, Sep 12 2015

			Triangle starts:
  3;
  1,  3;
  1,  4,  3;
  1,  5,  7,   3;
  1,  6, 12,  10,   3;
  1,  7, 18,  22,  13,   3;
  1,  8, 25,  40,  35,  16,   3;
  1,  9, 33,  65,  75,  51,  19,   3;
  1, 10, 42,  98, 140, 126,  70,  22,   3;
  1, 11, 52, 140, 238, 266, 196,  92,  25,   3;
  1, 12, 63, 192, 378, 504, 462, 288, 117,  28,  3;
  1, 13, 75, 255, 570, 882, 966, 750, 405, 145, 31, 3;

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Wolfdieter Lang, First 10 rows.

Row sums: A000079(n+1), n>=1, 3 if n=0. Alternating row sums are [3, -2, followed by 0's].
Column sequences (without leading zeros) give for m=1..9 with n>=0: A000027(n+3), A055998(n+1), A006503(n+1), A095661, A000574, A095662, A095663, A095664, A095665.
Cf. A097805.

a095660 n k = a095660_tabl !! n !! k
a095660_row n = a095660_tabl !! n
a095660_tabl = [3] : iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1,3]
-- Reinhard Zumkeller, Apr 08 2012

A095660:= func< n,k | n eq 0 select 3 else (1+2*k/n)*Binomial(n,k) >;
[A095660(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 02 2021

T(n,k):=piecewise(n=0,3,0Mircea Merca, Apr 08 2012

{3}~Join~Table[(1 + 2 k/n) Binomial[n, k], {n, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 14 2015 *)

def A095660(n,k): return 3 if n==0 else (1+2*k/n)*binomial(n,k)
flatten([[A095660(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

Recursion: T(n, m)=0 if m>n, T(0, 0)= 3; T(n, 0)=1 if n>=1; T(n, m) = T(n-1, m) + T(n-1, m-1).
G.f. column m (without leading zeros): (3-2*x)/(1-x)^(m+1), m>=0.
T(n,k) = (1+2*k/n) * binomial(n,k), for n>0. - Mircea Merca, Apr 08 2012
Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

A095662 Seventh column (m=6) of (1,3)-Pascal triangle A095660.

Original entry on oeis.org

3, 19, 70, 196, 462, 966, 1848, 3300, 5577, 9009, 14014, 21112, 30940, 44268, 62016, 85272, 115311, 153615, 201894, 262108, 336490, 427570, 538200, 671580, 831285, 1021293, 1246014, 1510320, 1819576, 2179672, 2597056, 3078768, 3632475

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 3-subset of an n-set X then, for n>=8, a(n-8) is the number of 6-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Sixth column: A000574. Eighth column: A095663.

CoefficientList[Series[(3-2x)/(1-x)^7,{x,0,40}],x] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{3,19,70,196,462,966,1848},40] (* Harvey P. Dale, Mar 30 2014 *)

G.f.: (3-2*x)/(1-x)^7.
a(n)= binomial(n+5, 5)*(n+18)/6 = 3*b(n)-2*b(n-1), with b(n):=binomial(n+6, 6); cf. A000579.
a(0)=3, a(1)=19, a(2)=70, a(3)=196, a(4)=462, a(5)=966, a(6)=1848, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Mar 30 2014

A095664 Ninth column (m=8) of (1,3)-Pascal triangle A095660.

Original entry on oeis.org

3, 25, 117, 405, 1155, 2871, 6435, 13299, 25740, 47190, 82654, 139230, 226746, 358530, 552330, 831402, 1225785, 1773783, 2523675, 3535675, 4884165, 6660225, 8974485, 11960325, 15777450, 20615868, 26700300, 34295052, 43709380, 55303380, 69494436, 86764260, 107666559, 132835365, 162994065

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 3-subset of an n-set X then, for n>=10, a(n-10) is the number of 8-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Eighth column: A095663. Tenth column: A095665.

x='x+O('x^66); Vec((3-2*x)/(1-x)^9) \\ Joerg Arndt, May 11 2013

a(n)= binomial(n+7, 7)*(n+24)/8 = 3*b(n)-2*b(n-1), with b(n):=binomial(n+8, 8); cf. A000581.

G.f.: (3-2*x)/(1-x)^9.

A095665 Tenth column (m=9) of (1,3)-Pascal triangle A095660.

Original entry on oeis.org

3, 28, 145, 550, 1705, 4576, 11011, 24310, 50050, 97240, 179894, 319124, 545870, 904400, 1456730, 2288132, 3513917, 5287700, 7811375, 11347050, 16231215, 22891440, 31865925, 43826250, 59603700, 80219568, 106919868, 141214920

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 3-subset of an n-set X then, for n>=11, a(n-11) is the number of 9-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Ninth column: A095663.

Table[Binomial[n+8,8] (n+27)/9,{n,0,30}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{3,28,145,550,1705,4576,11011,24310,50050,97240},30] (* Harvey P. Dale, Oct 13 2017 *)

a(n)= binomial(n+8, 8)*(n+27)/9 = 3*b(n)-2*b(n-1), with b(n):=binomial(n+9, 9); cf. A000582.

G.f.: (3-2*x)/(1-x)^10.

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A095660 Pascal (1,3) triangle.

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A095662 Seventh column (m=6) of (1,3)-Pascal triangle A095660.

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