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.

A176288 Hankel transform of A176287.

Original entry on oeis.org

1, 3, 15, 55, 131, 163, -169, -1521, -4437, -7429, -2945, 26471, 101587, 207699, 201639, -306497, -1907461, -4718165, -6464305, 547863, 30463779, 93816323, 161591287, 97035119, -400669877, -1676486565, -3504149217, -3693262649
Offset: 0

Views

Author

Paul Barry, Apr 14 2010

Keywords

Links

Programs

  • GAP
    a:=[1,3,15,55];; for n in [5..30] do a[n]:=6*a[n-1]-17*a[n-2]+24*a[n-3] -16*a[n-4]; od; a; # G. C. Greubel, Nov 25 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-3*x+14*x^2-8*x^3)/(1-3*x+4*x^2)^2 )); // G. C. Greubel, Nov 25 2019
  • Maple
    seq(coeff(series((1-3*x+14*x^2-8*x^3)/(1-3*x+4*x^2)^2, x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 25 2019
  • Mathematica
    LinearRecurrence[{6,-17,24,-16},{1,3,15,55},30] (* Harvey P. Dale, Jun 12 2017 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-3*x+14*x^2-8*x^3)/(1-3*x+4*x^2)^2) \\ G. C. Greubel, Nov 25 2019
  • Sage
    def A176288_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x+14*x^2-8*x^3)/(1-3*x+4*x^2)^2 ).list()
A176288_list(30) # G. C. Greubel, Nov 25 2019

Formula

G.f.: (1-3*x+14*x^2-8*x^3)/(1-3*x+4*x^2)^2.
a(n) = 2^n*( (2n+7)*sin(2n*atan(1/sqrt(7)))/sqrt(7) - (2*n-1)*cos(2n*atan(1/sqrt(7)))).

A092392 Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 20, 10, 4, 1, 70, 35, 15, 5, 1, 252, 126, 56, 21, 6, 1, 924, 462, 210, 84, 28, 7, 1, 3432, 1716, 792, 330, 120, 36, 8, 1, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1, 48620, 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1, 184756, 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1
Offset: 0

Views

Author

Ralf Stephan, Mar 21 2004

Keywords

Comments

First column is C(2*n,n) or A000984. Central coefficients are C(3*n,n) or A005809. - Paul Barry, Oct 14 2009
T(n,k) = A046899(n,n-k), k = 0..n-1. - Reinhard Zumkeller, Jul 27 2012
From Peter Bala, Nov 03 2015: (Start)
Viewed as the square array [binomial (2*n + k, n + k)]n,k>=0 this is the generalized Riordan array ( 1/sqrt(1 - 4*x),c(x) ) in the sense of the Bala link, where c(x) is the o.g.f. for A000108.
The square array factorizes as ( 1/(2 - c(x)),x*c(x) ) * ( 1/(1 - x),1/(1 - x) ), which equals the matrix product of A100100 with the square Pascal matrix [binomial (n + k,k)]n,k>=0. See the example below. (End)

Examples

			From _Paul Barry_, Oct 14 2009: (Start)
Triangle begins
  1,
  2, 1,
  6, 3, 1,
  20, 10, 4, 1,
  70, 35, 15, 5, 1,
  252, 126, 56, 21, 6, 1,
  924, 462, 210, 84, 28, 7, 1,
  3432, 1716, 792, 330, 120, 36, 8, 1
Production array is
  2, 1,
  2, 1, 1,
  2, 1, 1, 1,
  2, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
As a square array = A100100 * square Pascal matrix:
  /1   1  1  1 ...\   / 1          \/1 1  1  1 ...\
  |2   3  4  5 ...|   | 1 1        ||1 2  3  4 ...|
  |6  10 15 21 ...| = | 3 2 1      ||1 3  6 10 ...|
  |20 35 56 84 ...|   |10 6 3 1    ||1 4 10 20 ...|
  |70 ...         |   |35 ...      ||1 ...        |
- _Peter Bala_, Nov 03 2015

Links

Crossrefs

Rows 0-14 are A000984, A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.
Columns are A000217, A000292, A000332, A000389, A000579.
Diagonals are A005809, A045721, A025174, A004319, A013698, A003408.
Cf. A100100.

Programs

  • Haskell
    a092392 n k = a092392_tabl !! (n-1) !! (k-1)
a092392_row n = a092392_tabl !! (n-1)
a092392_tabl = map reverse a046899_tabl
-- Reinhard Zumkeller, Jul 27 2012
  • Magma
    /* As a triangle */ [[Binomial(2*n-k, n): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 22 2017
  • Maple
    A092392 := proc(n,k)
    binomial(2*n-k,n-k) ;
end proc:
seq(seq(A092392(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015
  • Mathematica
    Table[Binomial[2 n - k, n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 19 2016 *)
  • Maxima
    C(x):=(1-sqrt(1-4*x))/2;
A(x,y):=(1/sqrt(1-4*x))/(1-y*C(x));
taylor(A(x,y),y,0,10,x,0,10); /* Vladimir Kruchinin, Mar 19 2016 */
  • PARI
    for(n=0,10, for(k=0,n, print1(binomial(2*n - k,n), ", "))) \\ G. C. Greubel, Nov 22 2017

Formula

As a number triangle, this is T(n, k) = if(k <= n, C(2*n - k, n), 0). Its row sums are C(2*n + 1, n + 1) = A001700. Its diagonal sums are A176287. - Paul Barry, Apr 23 2005
G.f. of column k: 2^k/[sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))^k].
As a number triangle, this is the Riordan array (1/sqrt(1 - 4*x), x*c(x)), c(x) the g.f. of A000108. - Paul Barry, Jun 24 2005
G.f.: A(x,y)=1/sqrt(1 - 4*x)/(1-y*x*C(x)), where C(x) is g.f. of Catalan numbers. - Vladimir Kruchinin, Mar 19 2016

Extensions

Diagonal sums comment corrected by Paul Barry, Apr 14 2010
Offset corrected by R. J. Mathar, Feb 08 2013

A360211 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-3*k,n-2*k).

Original entry on oeis.org

1, 2, 5, 17, 61, 221, 812, 3021, 11344, 42899, 163146, 623320, 2390653, 9198879, 35494701, 137290466, 532149805, 2066501909, 8038146035, 31312535610, 122140123201, 477002869614, 1864912495716, 7298427590543, 28588888586743, 112080607196843, 439744801379594
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2023

Keywords

Crossrefs

Cf. A005317, A024718, A176332, A360185.
Cf. A000108, A176287.
Cf. A026641, A360212.

Programs

  • Maple
    A360211 := proc(n)
    add((-1)^k*binomial(2*n-3*k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360211(n),n=0..40) ; # R. J. Mathar, Mar 02 2023
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-3*k, n-2*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+2*x^2/(1+sqrt(1-4*x)))))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2 * c(x)) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence 2*n*a(n) +(-5*n+2)*a(n-1) +(-11*n+12)*a(n-2) +2*(-n+5)*a(n-3) +(-7*n+2)*a(n-4) +2*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k,n-2*k).

Original entry on oeis.org

1, 2, 7, 27, 107, 429, 1731, 7012, 28478, 115864, 471991, 1924483, 7852083, 32053208, 130893949, 534673600, 2184482707, 8926392419, 36479840422, 149095843951, 609400587426, 2490900041118, 10181669553847, 41618414303969, 170118507902985, 695366323719302
Offset: 0

Views

Author

Seiichi Manyama, Jan 28 2023

Keywords

Crossrefs

Cf. A014300, A106188, A108081, A114121, A176287.
Cf. A000108, A360144, A360150.

Programs

  • Maple
    A360149 := proc(n)
    add(binomial(2*n+k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360149(n),n=0..40) ; # R. J. Mathar, Mar 02 2023
  • Mathematica
    a[n_] := Sum[Binomial[2*n + k, n - 2*k], {k, 0, Floor[n/2]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(2*n+k, n-2*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^2*(2/(1+sqrt(1-4*x)))^5)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^2 * c(x)^5) ), where c(x) is the g.f. of A000108.
a(n) ~ sqrt((7 - 5*(2/(173 + 21*sqrt(69)))^(1/3) + ((173 + 21*sqrt(69))/2)^(1/3)) / 69) / ((4 - (2/(25 - 3*sqrt(69)))^(1/3) - ((25 - 3*sqrt(69))/2)^(1/3))/3)^n. - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence n*(47*n-1011)*a(n) +(-261*n^2 +8567*n -6378)*a(n-1) +2*(-165*n^2 -9388*n +16143)*a(n-2) +(3089*n^2 +919*n -27492)*a(n-3) +2*(-1283*n^2 +3900*n +3981)*a(n-4) +4*(81*n+11)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 02 2023
Showing 1-4 of 4 results.

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