A029609 -id:A029609 - cp's OEIS frontend

A095666 Pascal (1,4) triangle.

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

Offset: 0

Wolfdieter Lang, Jun 11 2004

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 (q=3), A096940 (q=5), A096956 (q=6).
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} 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

			Triangle begins:
  [4];
  [1,4];
  [1,5,4];
  [1,6,9,4];
  [1,7,15,13,4];
  ...

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

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.

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

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

A095666[n_, k_] := If[n == k,  4, (3*k/n + 1)*Binomial[n, k]];
Table[A095666[n, k], {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Apr 14 2025 *)

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

A146523 Binomial transform of A010685.

Original entry on oeis.org

1, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Philippe Deléham, Oct 30 2008

Linked to A029609 by a Catalan transform.

Hankel transform is (1, -15, 0, 0, 0, 0, 0, 0, 0, ...).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A010685, A020714, A029609, A084215, A109466.

CoefficientList[Series[(1+3x)/(1-2x), {x,0,50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 5*2^(Range[40] -1)] (* G. C. Greubel, Nov 23 2021 *)

a(n)=if(n,5<<(n-1),1) \\ Charles R Greathouse IV, Jan 17 2012

[1]+[5*2^(n-1) for n in (1..50)] # G. C. Greubel, Nov 23 2021

a(n) = 5*2^(n-1) for n >= 1, a(0) = 1.
a(n) = Sum_{k=0..n} A109466(n,k)*A029609(k).
a(n) = A084215(n+1) = A020714(n-1), n > 0. - R. J. Mathar, Nov 02 2008
G.f.: (1 + 3*x)/(1 - 2*x). - Vladimir Joseph Stephan Orlovsky, Jun 21 2011
G.f.: G(0), where G(k)= 1 + 3*x/(1 -  2*x/(2*x + 3*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
E.g.f.: (5*exp(2*x) - 3)/2. - Stefano Spezia, Feb 20 2023

A146533 Catalan transform of A135092.

Original entry on oeis.org

1, 7, 21, 70, 245, 882, 3234, 12012, 45045, 170170, 646646, 2469012, 9464546, 36402100, 140408100, 542911320, 2103781365, 8167621770, 31762973550, 123708423300, 482462850870, 1883902560540, 7364346373020, 28817007546600, 112866612890850, 442437122532132, 1735714865318364

Offset: 0

Philippe Deléham, Oct 31 2008

nonn

Cf. A088218, A000984, A029651, A100320, A029609, A144706.

a[n_]:=(7*Binomial[2n,n]-5*KroneckerDelta[n,0])/2; Array[a,27,0] (* Stefano Spezia, Feb 14 2025 *)

a(n) = Sum_{k=0..n} A039599(n,k)*A010687(k) = Sum_{k=0..n} A106566(n,k)*A135092(k).
a(n) = (7*C(2n,n) - 5*0^n)/2.
From Stefano Spezia, Feb 14 2025: (Start)
G.f.: (7/sqrt(1 - 4*x) - 5)/2.
E.g.f.: (7*exp(2*x)*BesselI(0, 2*x) - 5)/2. (End)

a(23)-a(26) from Stefano Spezia, Feb 14 2025

A146534 a(n) = 4C(2n,n) - 30^n.

Original entry on oeis.org

1, 8, 24, 80, 280, 1008, 3696, 13728, 51480, 194480, 739024, 2821728, 10816624, 41602400, 160466400, 620470080, 2404321560, 9334424880, 36300541200, 141381055200, 551386115280, 2153031497760, 8416395854880, 32933722910400, 128990414732400, 505642425751008, 1983674131792416

Offset: 0

Philippe Deléham, Oct 31 2008

nonn

Cf. A000984, A010688, A029609, A029651, A039599, A088218, A100320, A144706, A146533, A240530.

a[n_]:=4*Binomial[2n,n]-3*KroneckerDelta[n,0]; Array[a,27,0] (* Stefano Spezia, Feb 14 2025 *)

a(n) = Sum_{k=0..n} A039599(n,k)*A010688(k).
From Stefano Spezia, Feb 14 2025: (Start)
G.f.: 4/sqrt(1 - 4*x) - 3.
E.g.f.: 4*exp(2*x)*BesselI(0, 2*x) - 3. (End)

a(22)-a(26) from Stefano Spezia, Feb 14 2025

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A095666 Pascal (1,4) triangle.

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A146523 Binomial transform of A010685.

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A146533 Catalan transform of A135092.

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A146534 a(n) = 4*C(2n,n) - 3*0^n.

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A146534 a(n) = 4C(2n,n) - 30^n.