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

A122446 G.f. satisfies: A(x) = 1 + x*A(x)^2 + 2*x^2*(A(x)^2 - A(x)); equals the base sequence of pendular trinomial triangle A122445.

Original entry on oeis.org

1, 1, 2, 7, 24, 88, 336, 1321, 5316, 21788, 90640, 381750, 1624592, 6975136, 30177056, 131428917, 575765820, 2535433668, 11216757104, 49829385786, 222193501760, 994153952528, 4461915817760, 20082611971226, 90625360612296
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2006

Keywords

Comments

Functional equation for the g.f. is derived from the recurrence of the pendular triangle A122445. Iterated convolutions of this sequence with the central terms (A122447) generates all diagonals of A122445. For example: A122448 = A122446 * A122447; A122449 = A122446^2 * A122447.
Diagonal sums of triangle T with T(n,k) = 2^k*A133336(n,k). - Philippe Deléham, Nov 10 2009

Links

Crossrefs

Cf. A122445, A122447, A122448, A122449, A122450, A122451, A122452.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+2*x^2 -Sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) )); // G. C. Greubel, Mar 16 2021
  • Maple
    m:=30; S:=series( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 16 2021
  • Mathematica
    CoefficientList[Series[(1+2*x^2-Sqrt[1-4*x-4*x^2+4*x^4])/(2*x*(1+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 17 2013 *)
  • PARI
    {a(n)=polcoeff(2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))),n)}
  • Sage
    def A122446_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) ).list()
A122446_list(30) # G. C. Greubel, Mar 16 2021

Formula

G.f.: A(x) = (1 + 2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4))/(2*x*(1+2*x)).
Recurrence: (n+1)*a(n) = 2*(n-2)*a(n-1) + 12*(n-1)*a(n-2) + 8*(n-2)*a(n-3) - 4*(n-5)*a(n-4) - 8*(n-5)*a(n-5). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = (1/6)*(6+sqrt(6*(10 + 2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3))) + sqrt(6*(20-2^(2/3)*(43-3*sqrt(177))^(1/3) - 2^(2/3)*(43+3*sqrt(177))^(1/3) + 24*sqrt(6/(10+2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3)))))) = 4.797536514160165558... is the root of the equation 4 - 4*d^2 - 4*d^3 + d^4 = 0 and c = 0.908214882020417619380249683... is the positive root of the equation -59 - 944*c^2 - 2032*c^4 - 320*c^6 + 5184*c^8 = 0. - Vaclav Kotesovec, Sep 17 2013, updated Mar 18 2024

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
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