A119370 -id:A119370 - cp's OEIS frontend

A119369 Pendular trinomial triangle, read by rows of 2n+1 terms (n>=0), defined by the recurrence: if 0 < k < n, T(n,k) = T(n-1,k) + T(n,2n-1-k); otherwise, if n-1 < k < 2n-1, T(n,k) = T(n-1,k) + T(n,2n-2-k); with T(n,0)=T(n+1,2n)=1 and T(n+1,2n+1)=T(n+1,2n+2)=0.

1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 9, 7, 3, 1, 0, 0, 1, 4, 10, 20, 30, 23, 11, 4, 1, 0, 0, 1, 5, 15, 36, 70, 104, 81, 40, 16, 5, 1, 0, 0, 1, 6, 21, 58, 133, 253, 374, 293, 149, 63, 22, 6, 1, 0, 0, 1, 7, 28, 87, 226, 501, 938, 1380, 1087, 564, 248, 93, 29, 7, 1, 0, 0

Offset: 0

Paul D. Hanna, May 16 2006

The diagonals may be generated by iterated convolutions of a base sequence B with the sequence C of central terms. The g.f. B(x) of the base sequence satisfies: B = 1 + x*B^2 + x^2*(B^2 - B); the g.f. C(x) of the central terms satisfies: C(x) = 1/(1+x - x*B(x)).

			To obtain row 4, pendular sums of row 3 are carried out as follows.
  [1, 2, 3, 2, 1, 0, 0]: given row 3;
  [1, _, _, _, _, _, _]: start with T(4,0) = T(3,0) = 1;
  [1, _, _, _, _, _, 1]: T(4,6) = T(4,0) + T(3,6) = 1 + 0 = 1;
  [1, 3, _, _, _, _, 1]: T(4,1) = T(4,6) + T(3,1) = 1 + 2 = 3;
  [1, 3, _, _, _, 3, 1]: T(4,5) = T(4,1) + T(3,5) = 3 + 0 = 3;
  [1, 3, 6, _, _, 3, 1]: T(4,2) = T(4,5) + T(3,2) = 3 + 3 = 6;
  [1, 3, 6, _, 7, 3, 1]: T(4,4) = T(4,2) + T(3,4) = 6 + 1 = 7;
  [1, 3, 6, 9, 7, 3, 1]: T(4,3) = T(4,4) + T(3,3) = 7 + 2 = 9;
  [1, 3, 6, 9, 7, 3, 1, 0, 0]: complete row 4 by appending two zeros.
Triangle begins:
  1;
  1, 0,  0;
  1, 1,  1,  0,   0;
  1, 2,  3,  2,   1,   0,   0;
  1, 3,  6,  9,   7,   3,   1,    0,    0;
  1, 4, 10, 20,  30,  23,  11,    4,    1,   0,   0;
  1, 5, 15, 36,  70, 104,  81,   40,   16,   5,   1,  0,  0;
  1, 6, 21, 58, 133, 253, 374,  293,  149,  63,  22,  6,  1, 0, 0;
  1, 7, 28, 87, 226, 501, 938, 1380, 1087, 564, 248, 93, 29, 7, 1, 0, 0;
Central terms are:
  C = A119371 = [1, 0, 1, 2, 7, 23, 81, 293, 1087, 4110, ...].
Lower diagonals start:
  D1 = A119372 = [1, 1, 3, 9, 30, 104, 374, 1380, 5197, ...];
  D2 = A119373 = [1, 2, 6, 20, 70, 253, 938, 3546, 13617, ...].
  Diagonals above central terms (ignoring leading zeros) start:
  U1 = A119375 = [1, 3, 11, 40, 149, 564, 2166, 8420, ...];
  U2 = A119376 = [1, 4, 16, 63, 248, 980, 3894, 15563, ...].
There exists the base sequence:
  B = A119370 = [1, 1, 2, 6, 19, 64, 225, 816, 3031, 11473, ...]
which generates all diagonals by convolutions with central terms:
  D2 = B * D1 = B^2 * C
  U2 = B * U1 = B^2 * C"
where C" = [1, 2, 7, 23, 81, 293, 1087, ...]
are central terms not including the initial [1,0].

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A119370, A119371, A119372, A119373, A119374, A119375, A119376.

Variants: A118340, A118345, A118350.

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k<0 or k>2*(n-1) then 0
    elif n=2 and k<3 then 1
    else T(n-1, k) + `if`(kG. C. Greubel, Mar 16 2021

T[n_, k_]:= T[n, k]= If[n==0 && k==0, 1, If[k<0 || k>2*(n-1), 0, If[n==2 && k<3, 1, T[n-1, k] +If[kG. C. Greubel, Mar 16 2021 *)

T(n,k)= if(k==0 && n==0, 1, if(k>2*n-2 || k<0, 0, if(n==2 && k<=2, 1, T(n-1,k) + if(k

@CachedFunction
def T(n, k):
    if (n==0 and k==0): return 1
    elif (k<0 or k>2*(n-1)): return 0
    elif (n==2 and k<3): return 1
    else: return T(n-1, k) + ( T(n, 2*n-k-1) if kG. C. Greubel, Mar 16 2021

Sum_{k=0..2*n} T(n, k) = A119372(n). - G. C. Greubel, Mar 16 2021

A119371 G.f. satisfies: A(x) = (1+x) - x(4+x)A(x) + x(3+2x)*A(x)^2.

Original entry on oeis.org

1, 0, 1, 2, 7, 23, 81, 293, 1087, 4110, 15783, 61387, 241329, 957400, 3828055, 15410651, 62410871, 254095382, 1039394147, 4269718110, 17606507789, 72852976317, 302403773303, 1258855723796, 5254253027485, 21983753239135

Offset: 0

Paul D. Hanna, May 17 2006

nonn

Equals central terms of pendular trinomial triangle A119369.

			            A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 23*x^5 + 81*x^6 ...;
   -x*(4+x)*A(x) = -4*x -x^2 -4*x^3 -9*x^4 -30*x^5 -99*x^6 - ...;
x*(3+2*x)*A(x)^2 = 3*x +2*x^2 +6*x^3 +16*x^4 +53*x^5 +180*x^6 + ...;

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A119369, A119370, A119372, A119373, A119374, A119375, A119376.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+4*x+x^2 -Sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)) )); // G. C. Greubel, Mar 17 2021

m:= 30;
S:= series( (1+4*x+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 17 2021

CoefficientList[Series[(1+4*x+x^2-Sqrt[(1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)])/(2*x*(3+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 11 2013 *)

{a(n)=polcoeff((2*(1+x))/(1+4*x+x^2+sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x*O(x^n))),n)}

def A119371_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1+4*x+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)) ).list()
A119371_list(30) # G. C. Greubel, Mar 17 2021

G.f.: A(x) = (1+4*x+x^2 - sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x)))/(2*x*(3+2*x)).
G.f.: A(x) = 1/(1+x - x*B(x)) = (1 + x*H(x))/(1+x) = 1 + x^2*F(x)/B(x), where B(x) is g.f. of A119370, H(x) is g.f. of A119372, F(x) is g.f. of A119375.
Recurrence: 3*(n+1)*a(n) = 2*(5*n-4)*a(n-1) + 2*(7*n-8)*a(n-2) + 4*(n-2)*a(n-3) - 3*(n-5)*a(n-4) - 2*(n-5)*a(n-5). - Vaclav Kotesovec, Sep 11 2013
a(n) ~ sqrt(-z^2-3*z+1)*(4+2*z-z^3)^(n+1)*(35-8*z^3+12*z^2-2*z) /(242*sqrt(Pi)*n^(3/2)), where z = 1/(2*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) - 1/2*sqrt(8/3-1/3*(280-24*sqrt(129))^(1/3) - 2/3*(35+3*sqrt(129))^(1/3) + 8*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) = 0.225270426... is the root of the equation 1-2*z^2+z^4-4*z=0. - Vaclav Kotesovec, Sep 11 2013

A119372 G.f. satisfies: A(x) = 1 + x(1-x-x^2)A(x) + x^2(3+2x)*A(x)^2.

Original entry on oeis.org

1, 1, 3, 9, 30, 104, 374, 1380, 5197, 19893, 77170, 302716, 1198729, 4785455, 19238706, 77821522, 316506253, 1293489529, 5309112257, 21876225899, 90459484106, 375256749620, 1561259497099, 6513108751281, 27238006266620

Offset: 0

Paul D. Hanna, May 17 2006

nonn

Equals diagonal and row sums of pendular trinomial triangle A119369. Also equals convolution of A119370 and A119371 (central terms of A119369).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A119369, A119370, A119371, A119373, A119374, A119375, A119376.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-x+x^2+x^3 -(1+x)*Sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)) )); // G. C. Greubel, Mar 17 2021

m:= 30;
S:= series( (1-x+x^2+x^3 -(1+x)*sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 17 2021

CoefficientList[Series[(1-x+x^2+x^3-Sqrt[(1-x+x^2+x^3)^2-4*x^2*(3+2*x)])/(2*x^2*(3+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 11 2013 *)

{a(n)=polcoeff(2/(1-x+x^2+x^3+sqrt((1-x+x^2+x^3)^2-4*x^2*(3+2*x)+x*O(x^n))),n)}

def A119372_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-x+x^2+x^3 -(1+x)*sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)) ).list()
A119372_list(30) # G. C. Greubel, Mar 17 2021

G.f.: A(x) = (1-x+x^2+x^3 - sqrt( (1-x+x^2+x^3)^2 - 4*x^2*(3+2*x)) )/(2*x^2*(3+2*x)).
G.f.: A(x) = B(x)/(1+x - x*B(x)) = B(x)*G(x), where B(x) is g.f. of A119370, G(x) is g.f. of A119371.
Recurrence: 3*(n+2)*(2*n-1)*a(n) = (20*n^2 - 6*n - 11)*a(n-1) + (28*n^2 - 18*n + 5)*a(n-2) + (8*n^2-12*n-17)*a(n-3) - 3*(2*n^2 - 9*n + 1)*a(n-4) - 2*(n-5)*(2*n+1)*a(n-5). - Vaclav Kotesovec, Sep 11 2013
a(n) ~ sqrt(-8*z^2-5*z^3+2-5*z)*(4+2*z-z^3)^n*(-18-8*z+4*z^3+z^2)*(-35+8*z^3-12*z^2+2*z)/(242*sqrt(Pi)*n^(3/2)), where z = 1/(2*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) - 1/2*sqrt(8/3-1/3*(280-24*sqrt(129))^(1/3) - 2/3*(35+3*sqrt(129))^(1/3) + 8*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) = 0.225270426... is the root of the equation 1-2*z^2+z^4-4*z=0. - Vaclav Kotesovec, Sep 11 2013

A119373 A lower diagonal of pendular trinomial triangle A119369.

Original entry on oeis.org

1, 2, 6, 20, 70, 253, 938, 3546, 13617, 52967, 208255, 826315, 3304456, 13304924, 53891402, 219442686, 897772983, 3688451380, 15211545938, 62950542636, 261329456566, 1087985751336, 4541524025769, 19003488710465, 79696345430789

Offset: 0

Paul D. Hanna, May 17 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A119369, A119370, A119371, A119372, A119374, A119375, A119376.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( 8*(1+x)/( ((1+x^2) +Sqrt((1+x^2)^2 -4*x*(1+x)))^2*(1+4*x+x^2 +Sqrt((1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x))) ) )); // G. C. Greubel, Mar 16 2021

CoefficientList[Series[8*(1+x)/( ((1+x^2) + Sqrt[(1+x^2)^2 -4*x*(1+x)])^2*(1 + 4*x +x^2 +Sqrt[(1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x)])), {x,0,30}], x] (* G. C. Greubel, Mar 16 2021 *)

{a(n)=polcoeff(8*(1+x)/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x*O(x^n)))^2 /(1+4*x+x^2 + sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x)+x*O(x^n))),n)}

def A119373_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 8*(1+x)/( ((1+x^2) +sqrt((1+x^2)^2 -4*x*(1+x)))^2*(1+4*x+x^2 +sqrt((1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x))) ) ).list()
A119373_list(30) # G. C. Greubel, Mar 16 2021

G.f.: A(x) = B(x)^2/(1+x - x*B(x)) = B(x)^2*G(x) = B(x)*H(x), where B(x) is g.f. of A119370, G(x) is g.f. of A119371 and H(x) is g.f. of A119372.

G.f.: 8*(1+x)/( ((1+x^2) +sqrt((1+x^2)^2 -4*x*(1+x)))^2*(1+4*x+x^2 +sqrt((1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x))) ).

A119374 A lower diagonal of pendular trinomial triangle A119369.

Original entry on oeis.org

1, 3, 10, 36, 133, 501, 1918, 7440, 29180, 115522, 461044, 1852938, 7492846, 30464306, 124461782, 510696350, 2103708187, 8696498477, 36066269640, 150015248758, 625664295594, 2615929689642, 10962436020878, 46037427169060

Offset: 0

Paul D. Hanna, May 17 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A119369, A119370, A119371, A119372, A119373, A119375, A119376.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( 16*(1+x)/( ((1+x^2) +Sqrt((1+x^2)^2-4*x*(1+x)))^3*(1+4*x+x^2 +Sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x))) ) )); // G. C. Greubel, Mar 16 2021

CoefficientList[Series[16*(1+x)/( ((1+x^2) +Sqrt[(1+x^2)^2 -4*x*(1+x)])^3*(1+4*x +x^2 +Sqrt[(1+4*x+x^2)^2 -4*x*(1+x)*(3+2*x)])), {x,0,30}], x] (* G. C. Greubel, Mar 16 2021 *)

{a(n)=polcoeff(16*(1+x)/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x*O(x^n)))^3 /(1+4*x+x^2 + sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x)+x*O(x^n))),n)}

def A119374_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 16*(1+x)/( ((1+x^2) +sqrt((1+x^2)^2-4*x*(1+x)))^3*(1+4*x+x^2 +sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x))) ) ).list()
A119374_list(30) # G. C. Greubel, Mar 16 2021

G.f.: A(x) = B(x)^3/(1+x - x*B(x)) = B(x)^3*G(x) = B(x)^2*H(x) = B(x)*I(x), where B(x) is g.f. of A119370, G(x) is g.f. of A119371, H(x) is g.f. of A119372 and I(x) is g.f. of A119373.

G.f.: 16*(1+x)/( ((1+x^2) +sqrt((1+x^2)^2-4*x*(1+x)))^3*(1+4*x+x^2 +sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x))) ).

A119375 Diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.

Original entry on oeis.org

1, 3, 11, 40, 149, 564, 2166, 8420, 33074, 131085, 523599, 2105727, 8519469, 34652696, 141621164, 581266730, 2394961851, 9902433681, 41074316737, 170869972460, 712729001716, 2980264528670, 12490379959184, 52458339164169

Offset: 0

Paul D. Hanna, May 17 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A119369, A119370, A119371, A119372, A119373, A119374, A119376.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-2*x-x^2 - Sqrt(1-4*x-2*x^2+x^4))/( 1+2*x^3+x^4 +(1+x)^2*Sqrt(1-4*x-2*x^2+x^4) ) )); // G. C. Greubel, Mar 16 2021

CoefficientList[Series[(1-2*x-x^2 -Sqrt[1-4*x-2*x^2+x^4])/(x^2*(1+2*x^3+x^4 +(1+x)^2*Sqrt[1-4*x-2*x^2+x^4])), {x,0,30}], x] (* G. C. Greubel, Mar 16 2021 *)

{a(n)=polcoeff(2/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))* (2*(1+x)/(1+4*x+x^2 + sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x^3*O(x^n)))-1)/x^2,n)}

def A119375_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-2*x-x^2 -sqrt(1-4*x-2*x^2+x^4))/( x^2*(1+2*x^3+x^4 +(1+x)^2*sqrt(1-4*x-2*x^2+x^4)) ) ).list()
A119375_list(30) # G. C. Greubel, Mar 16 2021

G.f.: A(x) = B(x)*(G(x) - 1)/x^2 = B(x)*(B(x) - 1)/(x+x^2 - x^2*B(x)), where B(x) is g.f. of A119370 and G(x) is g.f. of A119371 (central terms of A119369).

G.f.: (1-2*x-x^2 -sqrt(1-4*x-2*x^2+x^4))/( x^2*(1+2*x^3+x^4 +(1+x)^2*sqrt(1-4*x-2*x^2+x^4)) ). - G. C. Greubel, Mar 16 2021

A119376 Second diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.

Original entry on oeis.org

1, 4, 16, 63, 248, 980, 3894, 15563, 62555, 252789, 1026623, 4188390, 17159382, 70570380, 291253664, 1205935204, 5008047097, 20854723702, 87064706122, 364334839028, 1527943938306, 6420911995109, 27033938458595

Offset: 0

Paul D. Hanna, May 17 2006

nonn

Equals convolution of A119370 and A119375, which is the prior diagonal above the central terms of triangle A119369.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A119369, A119370, A119371, A119372, A119373, A119374, A119375.

R:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-2*x^2+x^4) >;
Coefficients(R!( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) )); // G. C. Greubel, Mar 17 2021

f[x_]:= Sqrt[1-4*x-2*x^2+x^4];
CoefficientList[Series[2*(1-2*x-x^2 -f[x])/(x^2*(1+2*x^3+x^4 +(1+x)^2*f[x])*(1+x^2 +f[x])), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)

{a(n)=polcoeff(4/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))^2* (2*(1+x)/(1+4*x+x^2 + sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x^3*O(x^n)))-1)/x^2,n)}

def f(x): return sqrt(1-4*x-2*x^2+x^4)
def A119376_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) ).list()
A119376_list(30) # G. C. Greubel, Mar 17 2021

G.f.: A(x) = B(x)^2*(G(x) - 1)/x^2 = B(x)^2*(B(x) - 1)/(x+x^2 - x^2*B(x)), where B(x) is g.f. of A119370 and G(x) is g.f. of A119371 (central terms of A119369).

G.f.: 2*(1-2*x-x^2-f(x))/( x^2*(1+2*x^3+x^4+(1+x)^2*f(x))*(1+x^2+f(x)) ) where f(x) = sqrt(1-4*x-2*x^2+x^4). - G. C. Greubel, Mar 17 2021

A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 5, 1, 0, 14, 21, 9, 1, 0, 42, 84, 56, 14, 1, 0, 132, 330, 300, 120, 20, 1, 0, 429, 1287, 1485, 825, 225, 27, 1, 0, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 0, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 0, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1, 0

Offset: 0

Philippe Deléham, Oct 19 2007

Mirror image of triangle A086810; another version of A126216.
Equals A131198*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007
Diagonal sums: A119370. - Philippe Deléham, Nov 09 2009

			Triangle begins:
    1;
    1,    0;
    2,    1,    0;
    5,    5,    1,   0;
   14,   21,    9,   1,   0;
   42,   84,   56,  14,   1,  0;
  132,  330,  300, 120,  20,  1, 0;
  429, 1287, 1485, 825, 225, 27, 1, 0;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

[[Binomial(n-1,k)*Binomial(2*n-k,n)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[Binomial[n-1,k]*Binomial[2*n-k,n]/(n+1), {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 05 2018 *)

for(n=0,10, for(k=0,n, print1(binomial(n-1,k)*binomial(2*n-k,n)/(n+1), ", "))) \\ G. C. Greubel, Feb 05 2018

Sum_{k=0..n} T(n,k)*x^k = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - Philippe Deléham, Nov 02 2009

A104547 Number of Schroeder paths of length 2n having no UHD, UHHD, UHHHD, ..., where U=(1,1), D=(1,-1), H=(2,0).

Original entry on oeis.org

1, 2, 5, 16, 60, 245, 1051, 4660, 21174, 98072, 461330, 2197997, 10585173, 51443379, 251982793, 1242734592, 6165798680, 30754144182, 154123971932, 775669589436, 3918703613376, 19866054609754, 101029857327802, 515275408644773

Offset: 0

Emeric Deutsch, Mar 14 2005

nonn

A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).

Equals binomial transform of A119370. - Paul D. Hanna, May 17 2006

			a(2)=5 because we have HH, HUD, UDH, UDUD and UUDD (UHD does not qualify).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A006318, A104546, A119370.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-2*x+2*x^2 - Sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)) )); // G. C. Greubel, Jan 02 2023

CoefficientList[Series[(1-2*x+2*x^2 -Sqrt[1-8*x+16*x^2-12*x^3+4*x^4] )/(2*x*(1-x)), {x,0,40}], x] (* G. C. Greubel, Jan 02 2023 *)

{a(n)=polcoeff(2*(1-x)/(1-2*x+2*x^2 + sqrt(1-8*x+16*x^2-12*x^3+4*x^4+x*O(x^n))),n)} \\ Paul D. Hanna, May 17 2006

def A104547_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x+2*x^2 - sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)) ).list()
A104547_list(40) # G. C. Greubel, Jan 02 2023

a(n) = A104546(n, 0).
G.f.: G = G(z) satisfies G = 1 + z*G + z*G(G - z/(1-z)).
G.f.: (1-2*x+2*x^2 - sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)). - Paul D. Hanna, May 17 2006
D-finite with recurrence (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 12*(2*n-3)*a(n-2) + 2*(14*n-37)*a(n-3) - 2*(8*n-31)*a(n-4) + 4*(n-5)*a(n-5). - R. J. Mathar, Jul 26 2022

A364161 G.f. satisfies A(x) = 1 + xA(x)^2/(1 - x^3A(x)).

Original entry on oeis.org

1, 1, 2, 5, 15, 47, 153, 514, 1769, 6205, 22102, 79733, 290721, 1069688, 3966739, 14810348, 55627778, 210046102, 796864028, 3035912900, 11610468138, 44556451207, 171529074168, 662238211929, 2563524741603, 9947573055828, 38687704042595

Offset: 0

Seiichi Manyama, Aug 28 2023

nonn

Cf. A001003, A119370.

Cf. A218251, A364833, A365247.

A364161 := proc(n)
    add( binomial(n-2*k-1,k)*binomial(2*n-5*k+1,n-3*k)/(2*n5*k+1),k=0..floor(n/3)) ;
end proc:
seq(A364161(n),n=0..80); # R. J. Mathar, Aug 29 2023

a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*n-5*k+1, n-3*k)/(2*n-5*k+1));

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(2*n-5*k+1,n-3*k)/(2*n-5*k+1).

D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-2) +3*(-2*n+3)*a(n-3) +(-2*n+7)*a(n-5) +(n-8)*a(n-6) +(n-8)*a(n-8)=0. - R. J. Mathar, Aug 29 2023

cp's OEIS Frontend

A119369 Pendular trinomial triangle, read by rows of 2n+1 terms (n>=0), defined by the recurrence: if 0 < k < n, T(n,k) = T(n-1,k) + T(n,2n-1-k); otherwise, if n-1 < k < 2n-1, T(n,k) = T(n-1,k) + T(n,2n-2-k); with T(n,0)=T(n+1,2n)=1 and T(n+1,2n+1)=T(n+1,2n+2)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A119371 G.f. satisfies: A(x) = (1+x) - x*(4+x)*A(x) + x*(3+2*x)*A(x)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A119372 G.f. satisfies: A(x) = 1 + x*(1-x-x^2)*A(x) + x^2*(3+2*x)*A(x)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A119373 A lower diagonal of pendular trinomial triangle A119369.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A119374 A lower diagonal of pendular trinomial triangle A119369.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A119375 Diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A119371 G.f. satisfies: A(x) = (1+x) - x(4+x)A(x) + x(3+2x)*A(x)^2.

A119372 G.f. satisfies: A(x) = 1 + x(1-x-x^2)A(x) + x^2(3+2x)*A(x)^2.

A364161 G.f. satisfies A(x) = 1 + xA(x)^2/(1 - x^3A(x)).