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.

A176332 Row sums of triangle A176331.

Original entry on oeis.org

1, 2, 5, 16, 56, 202, 741, 2752, 10318, 38972, 148070, 565280, 2166646, 8332378, 32136205, 124249856, 481433286, 1868972828, 7267804550, 28304698336, 110383060776, 431000853028, 1684754608210, 6592277745536, 25818887839956
Offset: 0

Views

Author

Paul Barry, Apr 15 2010

Keywords

Comments

Hankel transform is A176333.
Let A(n) denote the n X n array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i-1 times to the tuple ((sqrt(-1))^m, 1 <= m <= n). Then the negative of the real part of the (n, n)-entry of A(n) equals a(n-2) for all n > 2. - John M. Campbell, Jan 20 2019

Links

Crossrefs

Cf. A000108, A176331, A176333.
Cf. A005317, A024718, A360185, A360211.

Programs

  • GAP
    List([0..30], n -> Sum([0..n], k -> Sum([0..n], j -> (-1)^(n-j)* Binomial(j, n-k)*Binomial(j, k) ))) # G. C. Greubel, Feb 22 2019
  • Magma
    [(&+[ (&+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]]): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 21 2019
  • Maple
    a:=n->add(add(binomial(j,n-k)*binomial(j,k)*(-1)^(n-j),j=0..n),k=0..n): seq(a(n),n=0..30); # Muniru A Asiru, Jan 23 2019
  • Mathematica
    f[n_]:= (-1)^n*Sum[Binomial[n+k, k] Cos[Pi(n+k)/2], {k, 0, n}]; Array[f, 24, 0] (* Robert G. Wilson v, Apr 02 2012 *)
  • PARI
    {a(n) = sum(k=0,n, sum(j=0,n, (-1)^(n-j)*binomial(j,n-k)* binomial(j,k))) };vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 21 2019
  • PARI
    a(n) = {my(v = vector(n, k, I^k)); for (k=1, n-1, v = vector(n, i, sum(j=1, i, v[j]));); -real(v[n]);} \\ Michel Marcus, Feb 25 2019
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k, n)); \\ Seiichi Manyama, Jan 29 2023
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^2*(2/(1+sqrt(1-4*x)))^2))) \\ Seiichi Manyama, Jan 29 2023
  • Sage
    [sum(sum((-1)^(n-j)*binomial(j,n-k)*binomial(j,k) for j in (0..n)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 21 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(j,n-k)*binomial(j,k)*(-1)^(n-j).
Logarithm g.f.: arctan(x*C(x)) = Sum_{n>=1} a(n)/n*x^n, where C(x) = (1-sqrt(1-4*x))/(2*x) (A000108). - Vladimir Kruchinin, Aug 10 2010
Conjecture: 6*n*a(n) + 2*(-17*n+10)*a(n-1) + (47*n-60)*a(n-2) + 10*(-3*n+5)*a(n-3) + 4*(2*n-5)*a(n-4) = 0. - R. J. Mathar, Nov 24 2012
Recurrence: 2*n*(5*n-8)*a(n) = 2*(25*n^2 - 50*n + 18)*a(n-1) - (45*n^2 - 92*n + 36)*a(n-2) + 2*(2*n-3)*(5*n-3)*a(n-3). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ 4^(n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2 * c(x)^2) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^2) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

A360186 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-6*k,n-3*k).

Original entry on oeis.org

1, 2, 6, 19, 68, 246, 905, 3364, 12624, 47715, 181392, 692808, 2656441, 10219208, 39423792, 152461079, 590861182, 2294182428, 8922674221, 34754402618, 135552346392, 529335200219, 2069344561102, 8097878381208, 31718268482881, 124341261876650
Offset: 0

Views

Author

Seiichi Manyama, Jan 29 2023

Keywords

Links

Crossrefs

Cf. A054108, A360185.
Cf. A000984, A360153.

Programs

  • Maple
    A360186 := proc(n)
    add((-1)^k*binomial(2*n-6*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360186(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    Table[Sum[(-1)^k Binomial[2n-6k,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Mar 05 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-6*k, n-3*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^3)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3) ).
a(n) ~ 2^(2*n + 6) / (65*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 29 2023
D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1) +n*a(n-3) +2*(-2*n+1)*a(n-4)=0. - R. J. Mathar, Mar 12 2023
a(n)+a(n-3) = A000984(n). - R. J. Mathar, Mar 12 2023

A360290 a(n) = Sum_{k=0..floor(n/2)} binomial(n-1-k,k) * binomial(2*n-4*k,n-2*k).

Original entry on oeis.org

1, 2, 6, 22, 82, 314, 1222, 4814, 19138, 76626, 308550, 1248230, 5069266, 20654602, 84392838, 345659166, 1418769154, 5834283298, 24031706246, 99134911542, 409495076050, 1693539077210, 7011618614342, 29058701620974, 120540377731266, 500443750830962
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2023

Keywords

Links

Crossrefs

Cf. A085362, A360291, A360292.
Cf. A360185, A360293, A383573.

Programs

  • Magma
    [&+[Binomial(n-1-k, k) * Binomial(2*n-4*k, n-2*k): k in [0..Floor(n div 2)]]: n in [0..30]]; // Vincenzo Librandi, May 04 2025
  • Mathematica
    Table[Sum[Binomial[n-1-k,k]* Binomial[2*n-4*k, n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 04 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(n-1-k, k)*binomial(2*n-4*k, n-2*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^2)))

Formula

G.f.: 1 / sqrt(1-4*x/(1-x^2)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(n-2)*a(n-2) - 2*(2*n-7)*a(n-3) - (n-4)*a(n-4).
a(n) ~ phi^(3*n) / (5^(1/4) * sqrt(Pi*n/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Feb 02 2023
a(n) = A383573(n) - A383573(n-2). - Seiichi Manyama, May 01 2025

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
Showing 1-4 of 4 results.

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