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 1-4 of 4 results.

A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 7, 15, 9, 1, 10, 36, 54, 27, 1, 13, 66, 162, 189, 81, 1, 16, 105, 360, 675, 648, 243, 1, 19, 153, 675, 1755, 2673, 2187, 729, 1, 22, 210, 1134, 3780, 7938, 10206, 7290, 2187, 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561, 1, 28, 351, 2592, 12474, 40824, 91854, 139968, 137781, 78732, 19683
Offset: 0

Views

Author

N. J. A. Sloane, May 03 2000

Keywords

Comments

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005
Triangle read by rows, n-th row = X^(n-1) * [1, 1, 0, 0, 0, ...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (3,3,3,...) in the subdiagonal; given row 0 = 1. - Gary W. Adamson, Jul 19 2008
Fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(2x+1)^n; see A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. - Clark Kimberling, Aug 04 2011

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  7,  15,   9;
  1, 10,  36,  54,   27;
  1, 13,  66, 162,  189,   81;
  1, 16, 105, 360,  675,  648,  243;
  1, 19, 153, 675, 1755, 2673, 2187, 729;

Links

Crossrefs

Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.
Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.
Cf. A084938, A098399, A110523, A133494, A136158, A193722.

Programs

  • Magma
    A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
[A038763(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 27 2023
  • Mathematica
    A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 27 2023 *)
  • PARI
    T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); T(n-1,k) + 3*T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jul 25 2023
  • SageMath
    def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 27 2023

Formula

T(n, 0)=1; T(1, 1)=1; T(n, k)=0 for k>n; T(n, k) = T(n-1, k-1)*3 + T(n-1, k) for n >= 2.
Sum_{k=0..n} T(n,k) = A081294(n). - Philippe Deléham, Sep 22 2006
T(n, k) = A136158(n, n-k). - Philippe Deléham, Dec 17 2007
G.f.: (1-2*x*y)/(1-(3*y+1)*x). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 27 2023: (Start)
T(n, 0) = A000012(n).
T(n, 1) = A016777(n-1).
T(n, 2) = A062741(n-1).
T(n, 3) = 9*A002411(n-2).
T(n, 4) = 27*A001296(n-3).
T(n, 5) = 81*A051836(n-4).
T(n, n) = A133494(n).
T(n, n-1) = A006234(n+2).
T(n, n-2) = A080420(n-2).
T(n, n-3) = A080421(n-3).
T(n, n-4) = A080422(n-4).
T(n, n-5) = A080423(n-5).
T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)

Extensions

More terms from Michel Marcus, Jul 25 2023

A098400 a(n) = 4^n*binomial(2*n+1, n).

Original entry on oeis.org

1, 12, 160, 2240, 32256, 473088, 7028736, 105431040, 1593180160, 24216338432, 369849532416, 5671026163712, 87246556364800, 1346089726771200, 20819521107394560, 322702577164615680, 5011381198321090560, 77954818640550297600, 1214454016715941478400
Offset: 0

Views

Author

Paul Barry, Sep 06 2004

Keywords

Links

Crossrefs

Cf. A000108, A001700, A069720, A069723, A098399, A151403.

Programs

  • Magma
    [4^n*(2*n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Dec 27 2023
  • Mathematica
    Table[4^n Binomial[2n+1,n],{n,0,20}] (* Harvey P. Dale, Jan 22 2019 *)
  • PARI
    a(n)=binomial(2*n+1,n)<<(2*n) \\ Charles R Greathouse IV, Oct 23 2023
  • SageMath
    [4^n*binomial(2*n+1,n) for n in range(31)] # G. C. Greubel, Dec 27 2023

Formula

G.f.: (1-sqrt(1-16*x))/(8*x*sqrt(1-16*x)).
E.g.f.: a(n) = n! * [x^n] exp(8*x)*(BesselI(0, 8*x) + BesselI(1, 8*x)). - Peter Luschny, Aug 25 2012
(n+1)*a(n) - 8*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 26 2012
a(n) = 4^n*(2*n+1)*Hypergeometric2F1([1-n,-n],[2],1). - Peter Luschny, Sep 22 2014
From G. C. Greubel, Dec 27 2023: (Start)
a(n) = 4^n * A001700(n).
a(n) = 4^n * (2*n+1) * A000108(n).
a(n) = (2*n+1) * A151403(n). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 8/15 + 128*arcsin(1/4)/(15*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 8/17 + 128*arcsinh(1/4)/(17*sqrt(17)). (End)

A164948 Fibonacci matrix read by antidiagonals. (Inverse of A136158.)

Original entry on oeis.org

1, 1, -1, 3, -4, 1, 9, -15, 7, -1, 27, -54, 36, -10, 1, 81, -189, 162, -66, 13, -1, 243, -648, 675, -360, 105, -16, 1, 729, -2187, 2673, -1755, 675, -153, 19, -1, 2187, -7290, 10206, -7938, 3780, -1134, 210, -22, 1, 6561, -24057, 37908, -34020, 19278, -7182, 1764, -276, 25, -1, 19683, -78732, 137781, -139968, 91854, -40824, 12474, -2592, 351, -28, 1
Offset: 0

Views

Author

Mark Dols, Sep 01 2009

Keywords

Comments

Triangle, read by rows, given by [1,2,0,0,0,0,0,0,0,...] DELTA [-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 02 2009

Examples

			As triangle:
    1;
    1,   -1;
    3,   -4,    1;
    9,  -15,    7,   -1;
   27,  -54,   36,  -10,    1;
   81, -189,  162,  -66,   13,   -1;
  243, -648,  675, -360,  105,  -16,    1;

Links

Crossrefs

Cf. A000007, A000326, A001296, A001519, A002411, A003688, A006234.
Cf. A016777, A038763, A080420, A080421, A081294, A084938, A098399.
Cf. A133494, A136158, A164942.

Programs

  • Magma
    A164948:= func< n,k | n eq 0 select 1 else (-1)^k*3^(n-k-1)*(n+2*k)*Binomial(n,k)/n >;
[A164948(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 26 2023
  • Mathematica
    A164948[n_,k_]:= If[n==0,1,(-1)^k*3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A164948[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2023 *)
  • SageMath
    def A164948(n,k): return 1 if (n==0) else (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A164948(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 26 2023

Formula

Sum_{k=0..n} T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001519(n).
From Philippe Deléham, Oct 09 2011: (Start)
T(n,k) = 3*T(n-1,k) - T(n-1,k-1) with T(0,0)=1, T(1,0)=1, T(1,1)=-1.
Row n: Expansion of (1-x)*(3-x)^(n-1), n>0. (End)
G.f.: (1-2*x)/(1-3*x+x*y). - R. J. Mathar, Aug 12 2015
From G. C. Greubel, Dec 26 2023: (Start)
T(n, k) = (-1)^k * A136158(n, k).
T(n, k) = (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k), for n > 0, with T(0, 0) = 1.
T(n, 0) = A133494(n).
T(n, 1) = -A006234(n+2), n >= 1.
T(n, 2) = A080420(n-2), n >= 2.
T(n, 3) = -A080421(n-3), n >= 3.
T(2*n, n) = 4*(-1)^n*A098399(n-1) - (1/3)*[n=0].
T(n, n-4) = 27*(-1)^n*A001296(n-3), n >= 4.
T(n, n-3) = 9*(-1)^(n-1)*A002411(n-2), n >= 3.
T(n, n-2) = 3*(-1)^n*A000326(n-1) = (-1)^n*A062741(n-1), n >= 2.
T(n, n-1) = (-1)^(n-1)*A016777(n-1), n >= 1.
T(n, n) = (-1)^n.
Sum_{k=0..n} (-1)^k*T(n, k) = A081294(n).
Sum_{k=0..n} (-1)^k*T(n-k, k) = A003688(n). (End)

Extensions

More terms from Philippe Deléham, Oct 09 2011

A098401 a(n) = (0^n + 3^n*binomial(2*n,n))/2.

Original entry on oeis.org

1, 3, 27, 270, 2835, 30618, 336798, 3752892, 42220035, 478493730, 5454828522, 62482581252, 718549684398, 8290957896900, 95938227092700, 1112883434275320, 12937269923450595, 150681143814306930, 1757946677833580850, 20540219077844997300, 240320563210786468410
Offset: 0

Views

Author

Paul Barry, Sep 06 2004

Keywords

Links

Crossrefs

Cf. A069723, A069720, A098399, A098400, A098402.

Programs

  • Magma
    [(0^n + 3^n * Binomial(2*n, n))/2: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012
  • Mathematica
    CoefficientList[Series[(6x)/(Sqrt[1-12x](1-Sqrt[1-12x])),{x,0,30}],x] (* Harvey P. Dale, Nov 29 2023 *)
Table[(3^n*Binomial[2*n,n] +Boole[n==0])/2, {n,0,40}] (* G. C. Greubel, Dec 27 2023 *)
a[n_] := 3^n*HypergeometricPFQ[{-n, -n + 1}, {1}, 1]; Flatten[Table[a[n], {n,0,20}]] (* Detlef Meya, May 21 2024 *)
  • SageMath
    [(3^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

Formula

a(n+1) = 3*A098399(n).
G.f.: 6*x/(sqrt(1-12*x)*(1-sqrt(1-12*x))).
n*a(n) - 6*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 13/11 + 24*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 11/13 - 24*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)
a(n) = 3^n*hypergeom([-n, -n + 1], [1], 1). - Detlef Meya, May 21 2024
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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