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.

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A082759 a(n) = Sum_{k = 0..n} binomial(n,k)*trinomial(n,k), where trinomial(n,k) = trinomial coefficients.

Original entry on oeis.org

1, 2, 8, 35, 160, 752, 3599, 17446, 85376, 420884, 2087008, 10398016, 52010479, 261021854, 1313707256, 6628095035, 33512880640, 169768235840, 861450392708, 4377796514152, 22277498220160, 113502759811000, 578931209245760, 2955873376166144, 15105883318474991
Offset: 0

Views

Author

Emanuele Munarini, May 21 2003

Keywords

Comments

Central coefficients of A115990. - Paul Barry, Feb 25 2011

Examples

			G.f. = 1 + 2*x + 8*x^2 + 35*x^3 + 160*x^4 + 752*x^5 + 3599*x^6 + 17446*x^7 + ...

Links

Crossrefs

Cf. A037011, A106228, A115990.

Programs

  • Maple
    a := n -> hypergeom([-n, -n, n + 1], [1/2, 1], 1/4):
seq(simplify(a(n)), n = 0..24);  # Peter Luschny, Jan 04 2025
  • Mathematica
    Table[Sum[Binomial[2 n - k, k] Binomial[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 24 2012; typo fixed by Vincenzo Librandi, May 07 2013 *)
  • PARI
    a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(n,k))

Formula

a(n) = Sum_{k = 0..n} C(n+k, n-k)*C(n, k). - Benoit Cloitre, Jun 20 2003
2*n*(2*n - 1)*(38*n - 53)*a(n) + ( - 760*n^3 + 1820*n^2 - 1252*n + 252)*a(n - 1) - 8*(n - 1)*(19*n^2 - 36*n + 9)*a(n - 2) - 3*(38*n - 15)*(n - 1)*(n - 2)*a(n - 3) = 0. - Vladeta Jovovic, Jul 15 2004
a(n) = Sum_{k = 0..n} C(2*n - k, k)*C(n, k). - Paul Barry, Jan 20 2005
a(n) ~ c * d^n / sqrt(Pi*n), where d = 5.21913624874158651... = (((1261 + 57*sqrt(57))^(2/3) + 112 + 10*(1261 + 57*sqrt(57))^(1/3))/(6*(1261 + 57*sqrt(57))^(1/3))) is the real root of the equation 4*d^3 - 20*d^2 - 4*d - 3 = 0 and c = 0.79036380822702870439029... = 1/114*sqrt(57)*sqrt((9747 + 57*sqrt(57))^(1/3)*(2*(9747 + 57*sqrt(57))^(2/3) + 912 + 57*(9747 + 57*sqrt(57))^(1/3)))/((9747 + 57*sqrt(57))^(1/3)) is the positive real root of the equation 1216*c^6 - 912*c^4 + 100*c^2 - 3 = 0. - Vaclav Kotesovec, Oct 24 2012 (updated Oct 16 2016, following a suggestion of Michael Somos)
G.f.: A(x) = x*B'(x)/B(x), where B(x) satisfies B(x) = x*(1 + 2*B(x) + 2*B(x)^2 + B(x)^3). - Vladimir Kruchinin, Jan 14 2015
a(n) = Sum_{k = 0..n} (-1)^k*C(n, k)*C(3*n - 2*k, n - k). - Peter Bala, Jul 13 2016
G.f. y = A(x) satisfies 0 = 1 + y*(3-2*x) + y^3*(-4+20*x+4*x^2+3*x^3). - Michael Somos, Oct 15 2016
From Peter Bala, Jan 09 2022: (Start)
a(n) = [x^n] (1 + 2*x + 2*x^2 + x^3)^n.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
a(n) = hypergeom([-n, -n, n + 1], [1/2, 1], 1/4). - Peter Luschny, Jan 04 2025

A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787333, 2290200, 6673662, 19478091, 56930961, 166613280, 488176938, 1431878079, 4203938697, 12353600427, 36331804089, 106932444885, 314946659951, 928213563878
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

a(n+1) is the number of Motzkin (2n)-paths whose last weak valley occurs immediately after step n. A weak valley in a Motzkin path (A001006) is an interior vertex whose following step has nonnegative slope and whose preceding step has nonpositive slope. For example, the weak valleys in the Motzkin path F.UF.FD.UD occur after the first, third and fifth steps as indicated by the dots (U=upstep of slope 1, D=downstep of slope -1, F=flatstep of slope 0) and, with n=2, a(3)=3 counts FFUD, UDUD, UFFD. - David Callan, Jun 07 2006
Starting with offset 2: (1, 3, 8, 22, 61, 171, 483, ...), = row sums of triangle A136537. - Gary W. Adamson, Jan 04 2008

Links

Crossrefs

First differences of A026135. Row sums of triangle A026105.
Pairwise sums of A005727. Column k=2 in A115990.
Cf. A001791, A132816.
Cf. A136537.

Programs

  • GAP
    List([0..30],i->Sum([0..Int(i/2)],k->Binomial(i-2,k)*Binomial(i-k,k))); # Muniru A Asiru, Mar 09 2019
  • Maple
    seq( sum('binomial(i-2,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
  • Mathematica
    CoefficientList[Series[x+(2x(x-1))/(1-3x-Sqrt[1-2x-3x^2]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2016 *)

Formula

G.f.: x + 2*x*(x-1)/(1-3x-sqrt(1-2x-3x^2)); for n > 1, first differences of the "directed animals" sequence A005773: a(n) = A005773(n) - A005773(n-1). - Emeric Deutsch, Aug 16 2002
Starting (1, 3, 8, 22, 61, 171, ...) gives the inverse binomial transform of A001791 starting (1, 4, 15, 56, 210, 792, ...). - Gary W. Adamson, Sep 01 2007
a(n) is the sum of the (n-2)-th row of triangle A131816. - Gary W. Adamson, Sep 01 2007
D-finite with recurrence n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Sep 15 2020

A362706 Number of squares formed by first n vertices of the infinite-dimensional hypercube.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 6, 7, 9, 13, 16, 21, 27, 36, 36, 37, 40, 45, 50, 57, 66, 78, 85, 94, 106, 121, 136, 154, 175, 200, 200, 201, 205, 211, 219, 229, 242, 258, 271, 286, 305, 327, 351, 378, 409, 444, 463, 484, 510, 539, 571, 606, 646, 690, 729, 771, 819
Offset: 1

Views

Author

Hugo van der Sanden, Jun 22 2023

Keywords

Comments

We can take the coordinates of a vertex to represent a binary number, so we define the n-th point to have coordinates represented by the binary expansion of n-1.
Let d(m) = a(m+1) - a(m) be the shifted first differences of a(n), so that d(m) represents the additional squares introduced by the (m+1)-th vertex. Then d(0) = d(2^x) = 0; when m = 2^x + 2^y, x > y, d(m) = A115990(x - 1, x - y - 1); generally, d(m) = sum d(k) for all k formed by selecting two 1's from the binary expansion of m. Thus d(7) = d(3) + d(5) + d(6).
a(n) is a lower bound for an infinite-dimensional extension of A051602. Peter Munn notes that it is not an upper bound: for example, the vertices of a regular {k-1}-simplex duplicated at unit distance in any orthogonal direction gives T_k squares from 2k+2 points, which exceeds a(n) at 6, 10 and 12 points.

Examples

			The 6 points (0,0,0), (1,0,0), (0,1,0), (1,1,0), (0,0,1), (1,0,1) give the squares (0,0,0), (1,0,0), (0,1,0), (1,1,0) and (0,0,0), (1,0,0), (0,0,1), (1,0,1). So a(6) = 2.

Crossrefs

Cf. A345340, A115990, A051602.

Formula

a(2^k) = A345340(k).
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