cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A122445 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) + 2*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.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 10, 8, 3, 1, 0, 0, 1, 4, 10, 22, 36, 28, 12, 4, 1, 0, 0, 1, 5, 15, 39, 83, 135, 107, 47, 17, 5, 1, 0, 0, 1, 6, 21, 62, 155, 324, 525, 418, 189, 72, 23, 6, 1, 0, 0, 1, 7, 28, 92, 259, 629, 1298, 2094, 1676, 773, 305, 104, 30, 7, 1, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Comments

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 + 2x^2*(B^2 - B); the g.f. C(x) of the central terms satisfies: C(x) = 1/(1+x - xB(x)).

Examples

			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) + 2*T(3,6) = 1 + 2*0 = 1;
  [1, 3, _, __, _, _, 1]: T(4,1) = T(4,6) + 1*T(3,1) = 1 + 1*2 = 3;
  [1, 3, _, __, _, 3, 1]: T(4,5) = T(4,1) + 2*T(3,5) = 3 + 2*0 = 3;
  [1, 3, 6, __, _, 3, 1]: T(4,2) = T(4,5) + 1*T(3,2) = 3 + 1*3 = 6;
  [1, 3, 6, __, 8, 3, 1]: T(4,4) = T(4,2) + 2*T(3,4) = 6 + 2*1 = 8;
  [1, 3, 6, 10, 8, 3, 1]: T(4,3) = T(4,4) + 1*T(3,3) = 8 + 1*2 = 10;
  [1, 3, 6, 10, 8, 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, 10,   8,   3,   1,   0,   0;
  1, 4, 10, 22,  36,  28,  12,   4,   1,  0,  0;
  1, 5, 15, 39,  83, 135, 107,  47,  17,  5,  1, 0, 0;
  1, 6, 21, 62, 155, 324, 525, 418, 189, 72, 23, 6, 1, 0, 0;
Central terms are:
  C = A122447 = [1, 0, 1, 2, 8, 28, 107, 418, 1676, 6848, ...].
Lower diagonals start:
  D1 = A122448 = [1, 1, 3, 10, 36, 135, 525, 2094, 8524, ...];
  D2 = A122449 = [1, 2, 6, 22, 83, 324, 1298, 5302, 22002, ...].
Diagonals above central terms (ignoring leading zeros) start:
  U1 = A122450 = [1, 3, 12, 47, 189, 773, 3208, 13478, 57222, ...];
  U2 = A122451 = [1, 4, 17, 72, 305, 1300, 5576, 24068, 104510, ...].
There exists the base sequence:
  B = A122446 = [1, 1, 2, 7, 24, 88, 336, 1321, 5316, 21788, ...]
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, 8, 28, 107, 418, 1676, 6848, 28418, ...]
are central terms not including the initial [1,0].

Links

Crossrefs

Cf. A122446, A122447 (central terms), A122452 (row sums).
Diagonals: A122448, A122449, A122450, A122451.
Variants: A118340, A118345, A118350, A119369.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {T(n,k)= if(k==0 && n==0, 1, if(k>2*n-2 || k<0, 0, if(n==2 && k<=2, 1, if(k
  • Sage
    @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

A122447 Central terms of pendular trinomial triangle A122445.

Original entry on oeis.org

1, 0, 1, 2, 8, 28, 107, 418, 1676, 6848, 28418, 119444, 507440, 2175500, 9400207, 40895602, 178984212, 787503168, 3481278734, 15454765948, 68871993872, 307981243608, 1381569997998, 6215433403188, 28036071086296
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Comments

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 A122446, H(x) is g.f. of A122448, F(x) is g.f. of A122450.

Links

Crossrefs

Cf. A122445, A122446, A122448, A122449, A122450, A122451, A122452.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) )); // G. C. Greubel, Mar 17 2021
  • Mathematica
    f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
CoefficientList[Series[(1+6*x+2*x^2-f[x])/(2*x*(4+3*x)), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)
  • PARI
    {a(n)=polcoeff(2*(1+2*x)/(1+6*x+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))),n)}
  • Sage
    def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
def A122447_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) ).list()
A122447_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f. satisfies: A(x) = 1+2*x - 2*x*(3+x)*A(x) + x*(4+3*x)*A(x)^2.
G.f.: A(x) = ( 1 +6*x +2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4) )/( 2*x*(4+3*x) ).

A122448 Diagonal elements A122445(n+1,n) of the pendular trinomial triangle A122445.

Original entry on oeis.org

1, 1, 3, 10, 36, 135, 525, 2094, 8524, 35266, 147862, 626884, 2682940, 11575707, 50295809, 219879814, 966487380, 4268781902, 18936044682, 84326759820, 376853237480, 1689551241606, 7597003401186, 34251504489484
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Links

Crossrefs

Cf. A122445, A122446, A122447, A122449, A122450, A122451, A122452.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( 2/(1-x+2*x^2+2*x^3 +(1+x)*f(x)) )); // G. C. Greubel, Mar 17 2021
  • Mathematica
    f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
CoefficientList[Series[2/(1-x+2*x^2+2*x^3 +(1+x)*f[x]), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)
  • PARI
    {a(n) =polcoeff(2/(1-x+2*x^2+2*x^3 +(1+x)*sqrt(1-4*x-4*x^2+4*x^4 +x*O(x^n) )), n)}
  • Sage
    def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
def A122447_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2/(1-x+2*x^2+2*x^3 +(1+x)*f(x)) ).list()
A122447_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f.: A(x) = B(x)/(1 +x -x*B(x) ) where B(x) is the g.f. of A122446.
G.f. satisfies: A(x) = 1 + x*(1-2*x-2*x^2)*A(x) + x^2*(4+3*x)*A(x)^2.
G.f.: A(x) = 2/(1 -x +2*x^2 +2*x^3 + (1+x)*sqrt(1 -4*x -4*x^2 +4*x^4)).
D-finite with recurrence 4*(n+2)*a(n) +(-9*n-2)*a(n-1) +(-41*n+34)*a(n-2) +2*(-20*n+39)*a(n-3) +4*(n-7)*a(n-4) +4*(7*n-36)*a(n-5) +12*(n-6)*a(n-6)=0. - R. J. Mathar, Feb 06 2025

A122449 Diagonal elements A122445(n+2,n) of the pendular trinomial triangle A122445.

Original entry on oeis.org

1, 2, 6, 22, 83, 324, 1298, 5302, 22002, 92488, 392996, 1685232, 7283511, 31694460, 138746706, 610601374, 2699835614, 11988069480, 53433418716, 238986495540, 1072250526558, 4824638825032, 21765895919444, 98433111857436
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Links

Crossrefs

Cf. A122445, A122446, A122447, A122448, A122450, A122451, A122452.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( 2/(1-2*x-2*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*f(x)) )); // G. C. Greubel, Mar 17 2021
  • Mathematica
    f[x_] := Sqrt[1 - 4*x - 4*x^2 + 4*x^4];
CoefficientList[Series[2/(1-2*x-2*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*f[x]), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)
  • PARI
    {a(n)=local(A,B=2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x^2*O(x^n)))); A=B^2/(1+x-x*B);polcoeff(A,n,x)}
  • Sage
    def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
def A122449_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2/(1-2*x-2*x^2-2*x^3+4*x^4+4*x^5 +(1+2*x^2+2*x^3)*f(x)) ).list()
A122449_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f.: A(x) = B(x)^2/(1+x -x*B(x)) where B(x) is the g.f. of A122446.
G.f.: 2/(1 -2*x -2*x^2 -2*x^3 +4*x^4 +4*x^5 +(1 +2*x^2 +2*x^3)*f(x)), where f(x) = sqrt(1 -4*x -4*x^2 +4*x^4). - G. C. Greubel, Mar 17 2021
D-finite with recurrence -4*(n+3)*(37*n-56)*a(n) +(33*n^2-357*n+1624)*a(n-1) +4*(547*n^2-620*n-554)*a(n-2) +4*(1142*n^2-2566*n-1613)*a(n-3) +16*(180*n^2-588*n+65)*a(n-4) +4*(-331*n^2+1937*n+1076)*a(n-5) +8*(-320*n^2+2107*n-617)*a(n-6) -48*(19*n-13)*(n-7)*a(n-7)=0. - R. J. Mathar, Feb 06 2025

A122450 Diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.

Original entry on oeis.org

1, 3, 12, 47, 189, 773, 3208, 13478, 57222, 245134, 1058348, 4600571, 20118753, 88450897, 390721560, 1733348234, 7719287578, 34497374034, 154659735720, 695397289078, 3135087583426, 14168892518258, 64181607367952
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Links

Crossrefs

Cf. A122445, A122446, A122447, A122448, A122449, A122451, A122452.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))) )); // G. C. Greubel, Mar 17 2021
  • Mathematica
    f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
CoefficientList[Series[2*(1-2*x^2-f[x])/(x*(1+2*x^2+f[x])*(1-x+2*x^2+2*x^3+(1+x)*f[x])), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)
  • PARI
    {a(n)=local(A,B=2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x^2*O(x^n)))); A=B*(B-1)/x/(1+x-x*B);polcoeff(A,n,x)}
  • Sage
    def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
def A122449_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))) ).list()
A122449_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f.: A(x) = B(x)*(B(x)-1)/(x*(1+x -x*B(x))) where B(x) is the g.f. of A122446.
G.f.: 2*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))*(1-x+2*x^2+2*x^3+(1+x)*f(x))), where f(x) = sqrt(1 -4*x -4*x^2 +4*x^4). - G. C. Greubel, Mar 17 2021

A122451 A diagonal above central terms of pendular trinomial triangle A122445, ignoring leading zeros.

Original entry on oeis.org

1, 4, 17, 72, 305, 1300, 5576, 24068, 104510, 456332, 2002675, 8829892, 39096653, 173781548, 775183764, 3469084436, 15571135682, 70084045640, 316242702258, 1430351652352, 6483550388522, 29448610671464, 134010580021152
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Links

Crossrefs

Cf. A122445, A122446, A122447, A122448, A122449, A122450, A122452.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(1-x+2*x^2+2*x^3+(1+x)*f(x))) )); // G. C. Greubel, Mar 17 2021
  • Mathematica
    f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
CoefficientList[Series[4*(1-2*x^2-f[x])/(x*(1+2*x^2+f[x])^2*(1-x+2*x^2+2*x^3+(1+x)*f[x])), {x,0,30}], x] (* G. C. Greubel, Mar 17 2021 *)
  • PARI
    {a(n)=local(A,B=2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x^2*O(x^n)))); A=B^2*(B-1)/x/(1+x-x*B);polcoeff(A,n,x)}
  • Sage
    def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
def A122449_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(1-x+2*x^2+2*x^3+(1+x)*f(x))) ).list()
A122449_list(30) # G. C. Greubel, Mar 17 2021

Formula

G.f.: A(x) = B(x)^2*(B(x)-1)/(x*(1+x - x*B(x))) where B(x) is the g.f. of A122446.
G.f.: 4*(1-2*x^2-f(x))/(x*(1+2*x^2+f(x))^2*(1-x+2*x^2+2*x^3+(1+x)*f(x))), where f(x) = sqrt(1 -4*x -4*x^2 +4*x^4). - G. C. Greubel, Mar 17 2021

A122452 Row sums of pendular triangle A122445.

Original entry on oeis.org

1, 1, 3, 9, 32, 118, 455, 1803, 7304, 30104, 125834, 532154, 2272728, 9788310, 42464315, 185394551, 813950824, 3591328136, 15916173734, 70819784774, 316254424144, 1416906860412, 6367136425862, 28690381745294
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Links

Crossrefs

Cf. A122445, A122446, A122447, A122448, A122449, A122450, A122451.

Programs

  • Maple
    T := proc(n, k) option remember;
if k=0 and n=0 then 1;
elif k<0 or 2*(n-1)G. C. Greubel, Mar 17 2021
  • Mathematica
    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 17 2021 *)
  • Sage
    @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 17 2021
Showing 1-7 of 7 results.

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