A126331 -id:A126331 - cp's OEIS frontend

A126791 Binomial matrix applied to A111418.

1, 4, 1, 17, 7, 1, 75, 39, 10, 1, 339, 202, 70, 13, 1, 1558, 1015, 425, 110, 16, 1, 7247, 5028, 2400, 771, 159, 19, 1, 34016, 24731, 12999, 4872, 1267, 217, 22, 1, 160795, 121208, 68600, 28882, 8890, 1940, 284, 25, 1, 764388, 593019, 355890, 164136

Offset: 0

Philippe Deléham, Mar 14 2007

Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 4*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1.
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
From R. J. Mathar, Mar 12 2013: (Start)
The matrix inverse starts
      1;
     -4,     1;
     11,    -7,     1;
    -29,    31,   -10,    1;
     76,  -115,    60,  -13,    1;
   -199,   390,  -285,   98,  -16,   1;
    521, -1254,  1185, -566,  145, -19,   1;
  -1364,  3893, -4524, 2785, -985, 201, -22, 1; ... (End)

			Triangle begins:
      1;
      4,     1;
     17,     7,     1;
     75,    39,    10,    1;
    339,   202,    70,   13,    1;
   1558,  1015,   425,  110,   16,   1;
   7247,  5028,  2400,  771,  159,  19,  1;
  34016, 24731, 12999, 4872, 1267, 217, 22, 1; ...
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  4, 1
  1, 3, 1
  0, 1, 3, 1
  0, 0, 1, 3, 1
  0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1 (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

A126791 := proc(n,k)
    if n=0 and k = 0 then
        1 ;
    elif k <0 or k>n then
        0;
    elif k= 0 then
        4*procname(n-1,0)+procname(n-1,1) ;
    else
        procname(n-1,k-1)+3*procname(n-1,k)+procname(n-1,k+1) ;
    end if;
end proc: # R. J. Mathar, Mar 12 2013
T := (n,k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1, -n+1, 3/2)): seq(seq(T(n,k),k=1..n),n=1..10); # Peter Luschny, May 13 2016

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 4, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A026378(m+n+1).
Sum_{k=0..n} T(n,k) = 5^n = A000351(n).
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + x )*(1 + 3*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

A126953 Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 33, 11, 3, 1, 110, 36, 12, 3, 1, 366, 122, 39, 13, 3, 1, 1220, 405, 135, 42, 14, 3, 1, 4065, 1355, 447, 149, 45, 15, 3, 1, 13550, 4512, 1504, 492, 164, 48, 16, 3, 1, 45162, 15054, 5004, 1668, 540, 180, 51, 17, 3, 1

Offset: 0

Philippe Deléham, Mar 19 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

Riordan array (2/(1-6x+sqrt(1-4*x^2)),x*c(x^2)) where c(x)= g.f. of the Catalan numbers A000108. - Philippe Deléham, Jun 01 2013

			Triangle begins:
     1;
     3,    1;
    10,    3,   1;
    33,   11,   3,   1;
   110,   36,  12,   3,  1;
   366,  122,  39,  13,  3,  1;
  1220,  405, 135,  42, 14,  3, 1;
  4065, 1355, 447, 149, 45, 15, 3, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 3, 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} T(n,k) = A127359(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A126931(m+n).
Sum_{k=0..n} T(n,k)*(-2*k+1) = 2^n. - Philippe Deléham, Mar 25 2007

A098409 Expansion of 1/(sqrt(1-3x)sqrt(1-7*x)).

Original entry on oeis.org

1, 5, 27, 155, 931, 5775, 36645, 236325, 1542195, 10153775, 67313377, 448691985, 3004182349, 20188647185, 136094684907, 919884469275, 6232016686995, 42305974804575, 287706424085745, 1959685788407025, 13367193276457881, 91295551930615005, 624255065007468207

Offset: 0

Paul Barry, Sep 07 2004

Binomial transform of A081671. 3rd binomial transform of A000984. Binomial transform is A098410.
Largest coefficient of (1+5*x+x^2)^n; row sums of triangle in A126331. - Philippe Deléham, Oct 02 2007
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in five colors. - N-E. Fahssi, Feb 05 2008
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have five colors. - N-E. Fahssi, Mar 31 2008
Diagonal of rational function 1/(1 - (x^2 + 5*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Isaac DeJager, Madeleine Naquin and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012 (the sequence b_{5,n}).
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Column 5 of A292627. Cf. A182401.

Table[SeriesCoefficient[1/(Sqrt[1-3*x]*Sqrt[1-7*x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
CoefficientList[Series[1/(Sqrt[1-3x] Sqrt[1-7x]),{x,0,30}],x] (* Harvey P. Dale, Jun 20 2015 *)

a(n):=coeff(expand((1+5*x+x^2)^n),x^n);
makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */

x='x+O('x^66); Vec(1/(sqrt(1-3*x)*sqrt(1-7*x))) \\ Joerg Arndt, May 11 2013

{a(n) = sum(k=0, n, 7^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 5^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

G.f.: 1/sqrt(1-10*x+21*x^2).
E.g.f.: exp(5x)*BesselI(0, 2x).
a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n,k)*binomial(2k,k). - Paul Barry, Mar 08 2005
a(n) = [x^n] (1+5*x+x^2)^n. - Emanuele Munarini, Apr 27 2012
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ 7^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012
a(n) = Sum_{k=0..n} 7^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
From Peter Bala, Jan 10 2022: (Start)
exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 5*x + 26*x^2 + 140*x^3 + 777*x^4 + ... is the o.g.f. of A182401.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
a(n) = (1/Pi) * Integral_{x = -1..1} (3 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (7 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A128973 Triangle formed by reading A038622 mod 2 .

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1

Offset: 0

Philippe Deléham, Apr 28 2007

Also triangle formed by reading triangles A089942, A124733, A126331, A126791, A126970 modulo 2 .

			Triangle begins:
1;
0, 1;
1, 1, 1;
1, 1, 0, 1;
1, 0, 0, 1, 1;
0, 1, 1, 0, 0, 1;
1, 0, 0, 1, 1, 1, 1;
0, 1, 1, 0, 1, 1, 0, 1;
1, 0, 0, 0, 0, 0, 0, 1, 1;

T(n,0)=A035263(n). Sum_{k, 0<=k<=n}T(n,k)*(-1)^k = (-1)^n .

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A126791 Binomial matrix applied to A111418.

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A126953 Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

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A098409 Expansion of 1/(sqrt(1-3*x)*sqrt(1-7*x)).

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A128973 Triangle formed by reading A038622 mod 2 .

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A098409 Expansion of 1/(sqrt(1-3x)sqrt(1-7*x)).