A029600 -id:A029600 - cp's OEIS frontend

A029618 Numbers in (3,2)-Pascal triangle (by row).

1, 3, 2, 3, 5, 2, 3, 8, 7, 2, 3, 11, 15, 9, 2, 3, 14, 26, 24, 11, 2, 3, 17, 40, 50, 35, 13, 2, 3, 20, 57, 90, 85, 48, 15, 2, 3, 23, 77, 147, 175, 133, 63, 17, 2, 3, 26, 100, 224, 322, 308, 196, 80, 19, 2, 3, 29, 126, 324, 546, 630, 504, 276, 99, 21, 2, 3, 32, 155, 450, 870

Offset: 0

Mohammad K. Azarian

Reverse of A029600. - Philippe Deléham, Nov 21 2006
Triangle T(n,k), read by rows, given by (3,-2,0,0,0,0,0,0,0,...) DELTA (2,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2011
Row n: expansion of (3+2x)*(1+x)^(n-1), n>0. - Philippe Deléham, Oct 10 2011
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013

			Triangle begins as:
  1;
  3,  2;
  3,  5,  2;
  3,  8,  7,  2;
  3, 11, 15,  9,  2;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A029600, A084938, A228196, A228576, A016789 (2nd column), A005449 (3rd column), A006002 (4th column).

T:= function(n,k)
    if n=0 and k=0 then return 1;
    elif k=0 then return 3;
    elif k=n then return 2;
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 12 2019

A029618 := proc(n,k)
    if k < 0 or k > n then
        0;
    elif  n = 0 then
        1;
    elif k=0 then
        3;
    elif k = n then
        2;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jul 08 2015

T[n_, k_]:= T[n, k]= If[n==0 && k==0, 1, If[k==0, 3, If[k==n, 2, T[n-1, k-1] + T[n-1, k] ]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 13 2019 *)

T(n,k) = if(n==0 && k==0, 1, if(k==0, 3, if(k==n, 2, T(n-1, k-1) + T(n-1, k) ))); \\ G. C. Greubel, Nov 12 2019

@CachedFunction
def T(n, k):
    if (n==0 and k==0): return 1
    elif (k==0): return 3
    elif (k==n): return 2
    else: return T(n-1,k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(n,0)=3, T(n,n)=2; n, k > 0. - Boris Putievskiy, Sep 04 2013

G.f.: (-1-x*y-2*x)/(-1+x*y+x). - R. J. Mathar, Aug 11 2015

A095666 Pascal (1,4) triangle.

Original entry on oeis.org

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

A051432 (Terms in A029617)/2.

Original entry on oeis.org

4, 13, 7, 20, 45, 10, 161, 112, 50, 13, 273, 162, 63, 588, 435, 225, 16, 1023, 660, 2178, 1683, 396, 111, 19, 3861, 507, 130, 8151, 4004, 637, 22, 30745, 25168, 16380, 8372, 3290, 960, 196, 25, 55913, 41548, 24752, 11662, 4250, 1156, 221, 116688, 97461

Offset: 0

James Sellers

nonn

A029617, A029600.

A051433 (Terms in A029605)/2.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 12, 13, 7, 1, 25, 20, 1, 24, 45, 10, 1, 1, 40, 98, 154, 161, 112, 50, 13, 1, 138, 252, 315, 273, 162, 63, 1, 60, 390, 567, 588, 435, 225, 16, 1, 957, 1155, 1023, 660, 1, 84, 319, 825, 2112, 2178, 1683, 396, 111, 19, 1, 403, 1144, 4290, 3861, 507

Offset: 0

James Sellers

Cf. A029605, A029600.

A051434 (Terms in A029607)/2.

Original entry on oeis.org

4, 12, 13, 7, 25, 20, 24, 45, 10, 40, 98, 154, 161, 112, 50, 13, 138, 252, 315, 273, 162, 63, 60, 390, 567, 588, 435, 225, 16, 957, 1155, 1023, 660, 84, 319, 825, 2112, 2178, 1683, 396, 111, 19, 403, 1144, 4290, 3861, 507, 130, 112, 1547, 6006, 8151, 4004

Offset: 0

James Sellers

nonn

A029607, A029600.

A051435 (Terms in A029613)/2.

Original entry on oeis.org

12, 24, 40, 98, 154, 138, 252, 60, 390, 567, 957, 84, 319, 825, 2112, 403, 1144, 112, 1547, 6006, 144, 740, 2660, 7098, 14560, 23452, 30030, 884, 3400, 9758, 21658, 38012, 53482, 180, 4284, 13158, 31416, 59670, 91494, 114257, 17442, 44574, 91086

Offset: 0

James Sellers

nonn

A029613, A029600.

A239310 Numbers of the form A001700(n)*k, n>=1, k>=2.

Original entry on oeis.org

6, 9, 12, 15, 18, 20, 21, 24, 27, 30, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 57, 60, 63, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 93, 96, 99, 100, 102, 105, 108, 110, 111, 114, 117, 120, 123, 126, 129, 130, 132, 135, 138, 140, 141, 144, 147

Offset: 1

Bob Selcoe, Mar 31 2016

Numbers that are central coefficients T(2k,k) k>=2 in (a,b)-Pascal triangles, where (a,b) represent boundary conditions; i.e., T(2k,k) = (a+b)*A001700(k-1).

			a(n)=50 appears because A001700(2)=10, so T(6,3)=50 in (1,4)- and (2,3)-Pascal triangles.

Cf. A001700.

Cf. A007318 (Pascal's triangle), A029600 ((2,3)-Pascal triangle), A095666 ((1,4)-Pascal triangle).

is(n)=my(k=1,t=3); while(n>=2*t, if(n%t==0, return(1)); k++; t=binomial(2*k+1, k+1)); 0 \\ Charles R Greathouse IV, Apr 04 2016

a(n) ~ kn, where k = 2.441823902640895564.... (This constant exists since A001700 grows exponentially.) - Charles R Greathouse IV, Apr 04 2016

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A029618 Numbers in (3,2)-Pascal triangle (by row).

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A095666 Pascal (1,4) triangle.

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A051432 (Terms in A029617)/2.

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A051433 (Terms in A029605)/2.

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A051434 (Terms in A029607)/2.

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A051435 (Terms in A029613)/2.

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A239310 Numbers of the form A001700(n)*k, n>=1, k>=2.

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