A126568 -id:A126568 - cp's OEIS frontend

A133158 Binomial transform of A126568, second binomial transform of A026641.

1, 3, 12, 57, 294, 1578, 8658, 48177, 270774, 1533450, 8736432, 50016090, 287497380, 1658174352, 9591422286, 55618701057, 323225066790, 1882009941570, 10976834700792, 64119701075886, 375057555388884, 2196539772794172, 12878508015774468

Offset: 0

Philippe Deléham, Oct 08 2007

nonn

The Hankel transform of this sequence is 3^n (see A000244).

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Row sums of triangle in A124575.

CoefficientList[Series[(1 + 3*Sqrt[-1 + 2*x] / Sqrt[-1 + 6*x])/(4 - 6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 02 2023 *)

Conjecture: 2*n*a(n) + (-19*n+12)*a(n-1) + 6*(8*n-11)*a(n-2) + 36*(-n+2)*a(n-3) = 0. - R. J. Mathar, Jun 30 2013

a(n) ~ 2^(n + 1/2) * 3^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Nov 02 2023

A110877 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0) = 1, T(n,k) = 0 if n= 1: T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1) with x = 3.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 6, 15, 7, 1, 21, 58, 37, 10, 1, 79, 232, 179, 68, 13, 1, 311, 954, 837, 396, 108, 16, 1, 1265, 4010, 3861, 2133, 736, 157, 19, 1, 5275, 17156, 17726, 10996, 4498, 1226, 215, 22, 1, 22431, 74469, 81330, 55212, 25716, 8391, 1893

Offset: 0

Philippe Deléham, Sep 19 2005

Similar to A064189 (x = 1) and to A039599 (x = 2).
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 by 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
Row sums yield A126568. - Philippe Deléham, Oct 10 2007
5^n = (n-th row terms) dot (first n+1 terms in the series (1, 4, 7, 10, ...)). Example for row 4: 5^4 = 625 = (21, 58, 37, 10, 1) dot (1, 4, 7, 10, 13) = (21 + 232 + 259 + 100 + 13). - Gary W. Adamson, Jun 15 2011
Riordan array (2/(1+x+sqrt(1-6*x+5*x^2)), (1-3*x-sqrt(1-6*x+5*x^2))/(2*x)). - Philippe Deléham, Mar 04 2013

			Triangle begins:
      1;
      1,     1;
      2,     4,     1;
      6,    15,     7,     1;
     21,    58,    37,    10,     1;
     79,   232,   179,    68,    13,    1;
    311,   954,   837,   396,   108,   16,    1;
   1265,  4010,  3861,  2133,   736,  157,   19,   1;
   5275, 17156, 17726, 10996,  4498, 1226,  215,  22,  1;
  22431, 74469, 81330, 55212, 25716, 8391, 1893, 282, 25, 1;
  ...
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  1, 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

Cf. A002212, A033321, A039599, A064189.

The inverse of A126126.

A110877 := proc(n,k)
    if k > n then
        0;
    elif n= 0 then
        1;
    elif k = 0 then
        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, Sep 06 2013

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, 1, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

T(n, 0) = A033321(n) and for k >= 1: T(n, k) = Sum_{j>=1} T(n-j, k-1)*A002212(j).
Sum_{k=0..n} T(m, k)*T(n, k) = T(m+n, 0) = A033321(m+n).
The triangle may also be generated from M^n * [1,0,0,0,...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and (1,3,3,3,...) in the main diagonal. - Gary W. Adamson, Dec 17 2006
Sum_{k=0..n} T(n,k)*(3*k+1) = 5^n. - Philippe Deléham, Feb 26 2007
Sum_{k=0..n} T(n,k) = A126568(n). - Philippe Deléham, Oct 10 2007

A126569 Top-left "head" entry of the n-th power of the E8 Cartan matrix.

Original entry on oeis.org

1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63442, 232828, 867146, 3269060, 12446684, 47771496, 184544427, 716658870, 2794956099, 10938266562, 42930256917, 168890693650, 665739119129, 2628578437646, 10393091551794, 41141896235012, 163028816478833

Offset: 0

Gary W. Adamson, Dec 28 2006

nonn

			a(6) = 430 = leftmost term in M^6 * [1,0,0,0,0,0,0,0].

Wikipedia, E8
Index entries for linear recurrences with constant coefficients, signature (16,-105,364,-714,784,-440,96,-1).

Cf. A126566, A126567, A126568.

E8 := matrix(8,8,[ [2, -1, 0, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0, -1 ], [ 0, 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, 0, -1, 2, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 2 ] ]) ;
printf("1,") ; for n from 1 to 20 do T := evalm(E8^n) ; printf("%a,", T[1,1]) ; od: # R. J. Mathar, May 08 2009

a(n) = leftmost term in M^n * [1,0,0,0,0,0,0,0], where M = the 8x8 matrix [2,-1,0,0,0,0,0,0; -1,2,-1,0,0,0,0,0; 0,-1,2,-1,0,0,0,-1; 0,0,-1,2,-1,0,0,0; 0,0,0,-1,2,-1,0,0; 0,0,0,0,-1,2,-1,0; 0,0,0,0,0,-1,2,0; 0,0,-1,0,0,0,0,2].
a(n) = 16*a(n-1)-105*a(n-2)+364*a(n-3)-714*a(n-4)+784*a(n-5)-440*a(n-6)+96*a(n-7) -a(n-8). - R. J. Mathar, May 08 2009 [Corrected by Georg Fischer, Mar 12 2020]
G.f.: -(2*x-1)*(2*x^2-4*x+1)*(x^4-16*x^3+20*x^2-8*x+1) / (1-16*x +105*x^2 -364*x^3+714*x^4-784*x^5+440*x^6-96*x^7+x^8). - R. J. Mathar, May 08 2009

Edited by R. J. Mathar, May 08 2009

A126567 Sequence generated from the E6 Cartan matrix.

Original entry on oeis.org

1, 2, 5, 14, 42, 132, 430, 1444, 4981, 17594, 63441, 232806, 866870, 3266460, 12426210, 47629020, 183638729, 711285170, 2764753405, 10775740030, 42086252770, 164635420788, 644811687734, 2527808259668, 9916569410301, 38923511495402, 152841133694345, 600349070362454

Offset: 0

Gary W. Adamson, Dec 28 2006

Wikipedia, E6 (mathematics)
Index entries for linear recurrences with constant coefficients, signature (12,-55,120,-125,52,-3).

Cf. A126566, A126568, A126569.

f[n_] := (MatrixPower[{{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, {0, -1, 2, -1, 0, -1}, {0, 0, -1, 2, -1, 0}, {0, 0, 0, -1, 2, 0}, {0, 0, -1, 0, 0, 2}}, n].{1, 0, 0, 0, 0, 0})[[1]]; Table[ f@n, {n, 0, 25}] (* Robert G. Wilson v, Aug 07 2007 *)

a(n) = ([2,-1,0,0,0,0; -1,2,-1,0,0,0; 0,-1,2,-1,0,-1; 0,0,-1,2,-1,0; 0,0,0,-1,2,0; 0,0,-1,0,0,2]^n)[1,1]; \\ Michel Marcus, Jan 30 2023

Let M = [2,-1,0,0,0,0; -1,2,-1,0,0,0; 0,-1,2,-1,0,-1; 0,0,-1,2,-1,0; 0,0,0,-1,2,0; 0,0,-1,0,0,2] then a(n) is the upper left term in M^n.
G.f.: -(2*x-1)*(2*x^4-16*x^3+20*x^2-8*x+1) / ((x-1)*(3*x-1)*(x^4-16*x^3+20*x^2-8*x+1)). - Colin Barker, May 25 2013
a(n) ~ c*(2 + sqrt(2 + sqrt(3)))^n, where c = (3 - sqrt(3))/24. - Stefano Spezia, Jan 29 2023
a(n) = (3^n + 1)/4 + ((3 + sqrt(3))*((2 - sqrt(2 - sqrt(3)))^n + (2 + sqrt(2 - sqrt(3)))^n) + (3 - sqrt(3))*((2 - sqrt(2 + sqrt(3)))^n + (2 + sqrt(2 + sqrt(3)))^n))/24. - Vaclav Kotesovec, Jan 30 2023

More terms from Robert G. Wilson v, Aug 07 2007

A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 21, 6, 1, 1, 93, 25, 7, 1, 1, 421, 112, 29, 8, 1, 1, 1937, 510, 132, 33, 9, 1, 1, 9017, 2357, 606, 153, 37, 10, 1, 1, 42349, 11009, 2819, 709, 175, 41, 11, 1, 1, 200277, 51840, 13233, 3324, 819, 198, 45, 12, 1, 1

Offset: 0

Philippe Deléham, Dec 06 2009

Expansion of row sums of T_(x,3), T_(x,y) defined in A039599.

Matrix product P^3 * Q * P^(-3), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A158815. - Peter Bala, Jul 13 2021

			Triangle begins:
    1;
    1,   1;
    5,   1,  1;
   21,   6,  1, 1;
   93,  25,  7, 1, 1;
  421, 112, 29, 8, 1, 1;
  ...

Cf. A061554, A158793, A158815, A171224.

Sum_{k=0..n} T(n,k)*x^k = A126952(n), A126568(n), A026375(n), A026378(n+1), A000351(n) for x = 0,1,2,3,4 respectively.

A171650 Triangle T, read by rows : T(n,k) = A007318(n,k)*A026641(n-k).

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 13, 12, 3, 1, 46, 52, 24, 4, 1, 166, 230, 130, 40, 5, 1, 610, 996, 690, 260, 60, 6, 1, 2269, 4270, 3486, 1610, 455, 84, 7, 1, 8518, 18152, 17080, 9296, 3220, 728, 112, 8, 1, 32206, 76662, 81684, 51240, 20916, 5796, 1092, 144, 9, 1

Offset: 0

Philippe Deléham, Dec 13 2009

			Triangle begins as
    1;
    1,   1;
    4,   2,   1;
   13,  12,   3,  1;
   46,  52,  24,  4, 1;
  166, 230, 130, 40, 5, 1; ...

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

[[(-1)^(n-k)*Binomial(n,k)*(&+[(-1)^j*Binomial(n-k+j,j): j in [0..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 29 2019

T[n_, k_]:= (-1)^(n-k)*Binomial[n, k]*Sum[(-1)^j*Binomial[n-k+j, j], {j, 0, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 29 2019 *)

{T(n,k) = (-1)^(n-k)*binomial(n,k)*sum(j=0,n-k,(-1)^j*binomial(n-k+j,j))}; \\ G. C. Greubel, Apr 29 2019

[[(-1)^(n-k)*binomial(n,k)*sum((-1)^j*binomial(n-k+j,j) for j in (0..n-k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 29 2019

Sum_{k, 0<=k<=n} T(n,k)*x^k = A127361(n), A127328(n), A026641(n), A126568(n), A133158(n) for x = -2, -1, 0, 1, 2 respectively.

T(n, k) = (-1)^(n-k)*binomial(n, k)*Sum_{j=0..n-k} (-1)^j*Binomial(n-k+j, j). - G. C. Greubel, Apr 29 2019

cp's OEIS Frontend

A133158 Binomial transform of A126568, second binomial transform of A026641.

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A110877 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0) = 1, T(n,k) = 0 if n= 1: T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1) with x = 3.

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A126569 Top-left "head" entry of the n-th power of the E8 Cartan matrix.

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A126567 Sequence generated from the E6 Cartan matrix.

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A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

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A171650 Triangle T, read by rows : T(n,k) = A007318(n,k)*A026641(n-k).

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