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 1-3 of 3 results.

A295113 a(n) = (1/n)*Sum_{k=0..n-1} (8*k + 9)*A295112(k)^2.

Original entry on oeis.org

9, 13, 17, 219, 1561, 8169, 39321, 191389, 985201, 5430789, 31943961, 198471183, 1288665177, 8665236121, 59922226809, 423935337411, 3056528058577, 22392246851973, 166311049602681, 1250027314777795, 9494339129623329, 72784922204637153, 562626619763553217, 4381665416129531961, 34354964747652467697
Offset: 1

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Author

Zhi-Wei Sun, Nov 14 2017

Keywords

Comments

Conjecture: a(n) is always integral for every n = 1,2,3,.... Moreover, for any odd prime p we have a(p) == 24 + 10*Leg(-1,p) - 18*Leg(3,p) - 9*Leg(p,3) (mod p^2), where Leg(m,p) denotes the Legendre symbol (m/p).
We also observe that Sum_{k=0}^{p-1}A295112(k)^2 == 2 (mod p) for any prime p > 3.

Examples

			a(2) = 13 since (1/2)*Sum_{k=0..1} (8k + 9)*A295112(k)^2 = (1/2)*((8*0 + 9)*A295112(0)^2 + (8 + 9)*A295112(1)^2) = (1/2)*(9*(-1)^2 + 17*(-1)^2) = 13.
		

Crossrefs

Programs

  • Mathematica
    W[n_]:=W[n]=Sum[Binomial[n,2k]Binomial[2k,k]/(2k-1),{k,0,n/2}];
    a[n_]:=a[n]=1/n*Sum[(8k+9)W[k]^2,{k,0,n-1}];
    Table[a[n],{n,1,25}]

A295132 a(n) = (2/n)*Sum_{k=1..n} (2k+1)*M(k)^2 where M(k) is the Motzkin number A001006(k).

Original entry on oeis.org

6, 23, 90, 432, 2286, 13176, 80418, 513764, 3400518, 23167311, 161640554, 1150633512, 8332048638, 61232315553, 455830692210, 3432015694314, 26101221114582, 200295455169015, 1549473966622602, 12074304397434552, 94713783502786686, 747454269790900728
Offset: 1

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Author

Zhi-Wei Sun, Nov 15 2017

Keywords

Comments

Sun (2014) conjectures that for any prime p > 3 we have Sum_{k = 0..p-1} M(k)^2 == (2 - 6*p)(p/3) (mod p^2) and Sum_{k = 0..p - 1} k*M(k)^2 == (9*p - 1)(p/3) (mod p^2), where (p/3) is the Legendre symbol.
Sun (2018) proves that a(n) is always an integer.

Examples

			a(2) = 23 since (2/2)*Sum_{k=1..2} (2k + 1)*M(k)^2 = (2*1 + 1)*M(1)^2 + (2*2 + 1)*M(2)^2 = 3*1^2 + 5*2^2 = 23.
		

Crossrefs

Programs

  • Maple
    h := k -> (4*k+2)*hypergeom([(1-k)/2,-k/2],[2],4)^2:
    a := proc(n) add(simplify(h(k)),k=1..n): if % mod n = 0 then %/n else -1 fi end:
    seq(a(n), n=1..25); # Peter Luschny, Nov 16 2017
  • Mathematica
    M[n_] := M[n] = Sum[Binomial[n, 2k] Binomial[2k, k]/(k + 1), {k, 0, n/2}];
    a[n_] := a[n] = 2/n * Sum[(2k + 1) M[k]^2, {k, 1, n}];
    Table[a[n], {n, 1, 25}]

Formula

a(n) = 2*A005043(n+1)*((6+6/n)*A005043(n) + (2+1/n)*A005043(n+1)). - Mark van Hoeij, Nov 10 2022

A385641 Partial sums of A097893.

Original entry on oeis.org

1, 3, 8, 20, 51, 133, 356, 972, 2695, 7557, 21372, 60840, 174097, 500295, 1442720, 4172752, 12099411, 35161001, 102375400, 298586652, 872177273, 2551118623, 7471195500, 21904500500, 64286141881, 188844619563, 555216323396, 1633658183432, 4810340397375, 14173698242137
Offset: 0

Views

Author

Mélika Tebni, Aug 03 2025

Keywords

Comments

Second partial sums of the central trinomial coefficients (A002426).
Third partial sums of A025178 (sequence starting 1, 0, 2, 4, 12, 32, 90 .... with offset 0).
For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).
For p Pythagorean prime (A002144), a(p) - 3 == 0 (mod p).
Sequences with g.f. (1-x)^k / sqrt(1-2*x-3*x^2): this sequence (k=-2), A097893 (k=-1), A002426 (k=0), A025178 (k=1), A024997 (k=2), A026083 (k=3). - Mélika Tebni, Aug 25 2025

Crossrefs

Programs

  • Maple
    a := series(exp(x)*(BesselI(0, 2*x) + 2*int(BesselI(0, 2*x), x) + int(int(BesselI(0, 2*x), x), x)), x = 0, 30): seq(n!*coeff(a, x, n), n = 0 .. 29);
  • PARI
    a(n) = sum(k=0, n, sum(i=0, k, sum(j=0, i, binomial(i, i-j)*binomial(j, i-j)))); \\ Michel Marcus, Aug 06 2025
  • Python
    from math import comb as C
    def a(n):
        return sum(C(n+1, k+1)*C(2*(k//2), k//2) for k in range(n + 1))
    print([a(n) for n in range(30)])
    

Formula

G.f.: (1 / sqrt((1 + x)*(1 - 3*x))) / (1 - x)^2.
E.g.f.: exp(x)*(BesselI(0, 2*x) + 2*g(x) + Integral_{x=-oo..oo} g(x) dx) where g(x) = Integral_{x=-oo..oo} BesselI(0, 2*x) dx.
D-finite with recurrence n*a(n) = (4*n-1)*a(n-1) - (2*n+1)*a(n-2) - (4*n-5)*a(n-3) + 3*(n-1)*a(n-4).
a(0) = 1, a(1) = 3 and a(n) = a(n-2) - 1 + 2*A383527(n) for n >= 2.
a(n) = Sum_{k=0..n} binomial(n+1, k+1)*A128014(k).
a(n) = Sum_{k=0..n} (2*A247287(k) + k+1).
a(n) ~ 3^(n + 5/2) / (8*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 03 2025
Sum_{k=0..n} A295112(n-k)*a(k) + binomial(n+3, 3) = 0. - Mélika Tebni, Sep 03 2025
Showing 1-3 of 3 results.