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.

A203577 Exponential (or binomial) half-convolution of the sequence A000108 (Catalan) with itself.

Original entry on oeis.org

1, 1, 4, 11, 58, 212, 1304, 5567, 37734, 178148, 1284124, 6501420, 48758648, 259775440, 2000594288, 11080668871, 86930955662, 496461841956, 3947716126292, 23113333523180, 185660199980696, 1109722749130576, 8983793097101144, 54645629076275356, 445109373450545608, 2748480598104423952
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2012

Keywords

Comments

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see A203576, where also the rule for the e.g.f. is given.

Examples

			With Catalan = A000108 = {1, 1, 2, 5, 14, 42, ...}
a(4) = 1*1*14 + 4*1*5 + 6*2*2 = 58.
a(5) = 1*1*42 + 5*1*14 + 10*2*5 = 212.

Links

Crossrefs

Cf. A203576, A000108, A014330 (exponential convolution).

Programs

  • Mathematica
    a[n_] := Sum[ Binomial[n, k]*CatalanNumber[k]*CatalanNumber[n - k], {k, 0, n/2}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 21 2013 *)
  • PARI
    hat(b,n) = sum(k=0,n\2,binomial(n,k)*b(k)*b(n-k))
A203577(n)=hat(A000108,n)  \\ where A000108(n)=(2*n)!/n!/(n+1)! \\ - M. F. Hasler, Jan 13 2012

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*Catalan(k)*Catalan(n-k), n >= 0.
E.g.f.: (C(x)^2 + C2(x^2))/2 with the e.g.f. C(x) of A000108, and the e.g.f. C2(x) := Sum_{n>=0} Catalan(n)^2*x^n/(n!)^2 of the scaled Catalan squares. See a comment above.
C(x) = hypergeom([1/2],[2],4*x) (see A000108 for the version involving BesselI functions), and
C2(x) = hypergeom([1/2,1/2],[1,2,2],16*x).
Recurrence: n*(n+1)^2 * (n+2)^2 * (3*n^6 - 39*n^5 + 166*n^4 - 322*n^3 + 316*n^2 - 153*n + 27)*a(n) = 12*(n-1)*n*(n+1)^2 * (3*n^7 - 34*n^6 + 113*n^5 - 121*n^4 - 19*n^3 + 68*n^2 + 17*n - 18)*a(n-1) + 32*(3*n^11 - 45*n^10 + 220*n^9 - 448*n^8 + 173*n^7 + 920*n^6 - 1696*n^5 + 842*n^4 + 580*n^3 - 846*n^2 + 360*n - 54)*a(n-2) - 768*(n-2)^3 * n *(3*n^7 - 34*n^6 + 113*n^5 - 121*n^4 - 19*n^3 + 68*n^2 + 17*n - 18)*a(n-3) + 2048*(n-3)^3 * (n-2)^2 * (3*n^6 - 21*n^5 + 16*n^4 + 12*n^3 + n^2 - 2)*a(n-4). - Vaclav Kotesovec, Feb 25 2014
a(n) ~ 2^(3*n+2)/(Pi*n^3) * (1 + (1+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, Feb 25 2014

A203578 Exponential (or binomial) half-convolution of A000045 (Fibonacci) with itself.

Original entry on oeis.org

0, 0, 2, 3, 14, 35, 155, 371, 1518, 3891, 15745, 40755, 161459, 426803, 1671175, 4469555, 17301630, 46805811, 179569163, 490156851, 1865624365, 5132989235, 19404565567, 53753361203, 201986220339, 562912506675, 2103942223775, 5894896300851, 21927151270703, 61732155503411
Offset: 0

Views

Author

Wolfdieter Lang, Jan 13 2012

Keywords

Comments

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see a comment on A203576 where also the rule for the e.g.f. is given.

Links

Crossrefs

Cf. A000045, 2*A014335 (exponential convolution), A032443.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]Fibonacci[k]Fibonacci[n-k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Mar 04 2013 *)

Formula

a(n) = sum(binomial(n,k)*F(k)*F(n-k),k=0..floor(n/2)), n>=0, with F(n)=A000045(n).
E.g.f.: (f(x)^2 + Fs2(x^2))/2, with the e.g.f. f(x) of A000045 and the o.g.f. Fs2(x):=sum((F(n)/n!)^2*x^n,n=0..infty) of the scaled squares. f(x)^2 = 2*exp(x)*(cosh((2*phi-1)*x)-1)/5 (see A000045 for f(x)) and Fs2(x^2) = (BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) - 2*BesselI(0,2*i*x))/5, with the golden section phi:=(1+sqrt(5))/2, the complex unit i, and for BesselI see Abramowitz-Stegun (reference and link given in A008277, p. 375, eq. 9.6.10). BesselI(0,2*sqrt(y)) = hypergeom([],[1],y) is the e.g.f. of the sequence {1/n!}.
Bisection:
a(2*k) = (A032443(k)*L(2*k) - (1 + (-1)^k*binomial(2*k,k)))/5 and a(2*k) = (2^(2*k)*L(2*k+1) - 1)/5, k>=0, with the Lucas numbers L(n)=A000032(n), and A032443(k)=(2^(2*k) + binomial(2*k,k))/2. - Wolfdieter Lang, Jan 16 2012.

A204449 Exponential (or binomial) half-convolution of A000032 (Lucas) with itself.

Original entry on oeis.org

4, 2, 8, 17, 84, 177, 737, 1857, 7732, 19457, 78223, 203777, 809145, 2134017, 8349013, 22347777, 86533892, 234029057, 897748577, 2450784257, 9328491339, 25664946177, 97021416973, 268766806017, 1009936510009, 2814562533377
Offset: 0

Views

Author

Wolfdieter Lang, Jan 16 2012

Keywords

Comments

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see A203576. There the rule for the e.g.f. is also found.
The other half of this exponential half-convolution is found under A204450.

Examples

			With A000032 = {2, 1, 3, 4, 7, 11,...}
  a(4) = 1*2*7 + 4*1*4 + 6*3*3 = 84,
  a(5) = 1*2*11 + 5*1*7 + 10*3*4 = 177.

Crossrefs

Cf. A000032, 2*A203579 (exponential convolution), A204450.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*LucasL[k]*LucasL[n-k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 25 2019 *)

Formula

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..floor(n/2)), n>=0, with L(n)=A000032(n).
E.g.f.: (l(x)^2 + L2(x^2))/2 with the e.g.f. l(x) of A000032, and the o.g.f. L2(x) of the sequence {(L(n)/n!)^2}.
l(x)^2 = 2*exp(x)*(cosh(sqrt(5)*x)+1) (see 2*A203579).
L2(x^2) = BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) + 2*BesselI(0,2*I*x), with the golden section phi:=(1+sqrt(5))/2, and for BesselI see Abramowitz-Stegun (reference and link given under A008277), p. 375, eq. 9.6.10.
BesselI(0,2*sqrt(x)) = hypergeom([],[1],x) is the e.g.f. of {1/n!}.
Bisection: a(2*k) = (2^(2*k)+binomial(2*k,k))*L(2*k)/2 +1 + ((-1)^k)*binomial(2*k,k), a(2*k+1) = 2^(2*k)*L(2*k+1)+1, k>=0. For (2^(2*k)+binomial(2*k,k))/2 see A032443(k).
Showing 1-3 of 3 results.

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