A107230 -id:A107230 - cp's OEIS frontend

A107238 A Chebyshev transform of number triangle A107230.

1, 1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 8, 4, 1, 0, 5, 15, 15, 5, 1, 0, 6, 27, 36, 24, 6, 1, 0, 7, 42, 84, 70, 35, 7, 1, 0, 8, 64, 160, 200, 120, 48, 8, 1, 0, 9, 90, 300, 450, 405, 189, 63, 9, 1, 0, 10, 125, 500, 1000, 1050, 735, 280, 80, 10, 1, 0, 11, 165, 825, 1925, 2695, 2156, 1232, 396

Offset: 0

Paul Barry, May 14 2005

Product of the number triangle A107230 by the Riordan array ((1-x^2)/(1+x^2),x/(1+x^2)). First column if C(1,n), second column is n (A001477), third column is essentially A034828.

			Triangle begins
1;
1,1;
0,2,1;
0,3,3,1;
0,4,8,4,1;
0,5,15,15,5,1;

Number triangle T(n, k)=sum{j=0..floor(n/2), (n/(n-j))(-1)^j*C(n-j, j)*A107230(n-2j, k)} (with T(0, n)=0^n).

A132894 Number of (1,0) steps in all paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., in all length-n left factors of Motzkin paths).

Original entry on oeis.org

0, 1, 4, 15, 52, 175, 576, 1869, 6000, 19107, 60460, 190333, 596652, 1863745, 5804176, 18028755, 55873872, 172818243, 533589660, 1644921789, 5063762220, 15568666029, 47811348816, 146675181975, 449538774048, 1376564658525

Offset: 0

Emeric Deutsch, Oct 07 2007

nonn

Number of peaks (i.e., UDs) in all paths of length n+1 with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., in all length n+1 left factors of Motzkin paths). Example: a(2)=4 because in the 13 (=A005773(4)) length-3 left factors of Motzkin paths, namely HHH, HHU, H(UD), HUH, HUU, (UD)H, (UD)U, UHD, UHH, UHU, U(UD), UUH and UUU, we have altogether 4 peaks (shown between parentheses).

This could be called the Motzkin transform of A077043 because the substitution x -> x*A001006(x) in the independent variable of the g.f. of A077043 yields the g.f. of this sequence here. - R. J. Mathar, Nov 10 2008

			a(2) = 4 because in the 5 (=A005773(3)) length-2 left factors of Motzkin paths, namely HH, HU, UD, UH and UU, we have altogether 4 H steps.
G.f. = x + 4*x^2 + 15*x^3 + 52*x^4 + 175*x^5 + 576*x^6 + 1869*x^7 + 6000*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. Asinowski and G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792, 2012; and also, J. Int. Seq. 17 (2014) #14.1.5.
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.

Cf. A005773, A107230, A132893.

Column k=1 of A328347.

a := n -> add(k*binomial(n, k)*binomial(n-k, floor((n-k)/2)), k=0..n): seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) a(n):=`if`(n<2, n, 2*n/(n-1)*a(n-1)+3*a(n-2)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 15 2013

a[n_] := n*Hypergeometric2F1[3/2, 1-n, 2, 4]; Table[ a[n] // Abs, {n, 0, 25}] (* Jean-François Alcover, Jul 10 2013 *)
a[ n_] := If[ n < 0, 0, -(-1)^n n Hypergeometric2F1[ 3/2, 1 - n, 2, 4]]; (* Michael Somos, Aug 06 2014 *)

A132894 = lambda n: (-1)^(n+1)*jacobi_P(n-1,1,-n+1/2,-7)
[Integer(A132894(n).n(40),16) for n in range(26)] # Peter Luschny, Sep 23 2014

a(n) = Sum_{k=0..n} k*A107230(n,k).
a(n) = Sum_{k=0..floor((n+1)/2)} k*A132893(n+1,k).
a(n) = Sum_{k=0..n} k*C(n,k)*C(n-k, floor((n-k)/2)).
G.f.: z/((1-3*z)*sqrt(1-2*z-3*z^2)).
a(n) = Sum_{k=0..n} k*C(n,k)*C(2*k,k)*(-1)^(n-k). - Wadim Zudilin, Oct 11 2010
E.g.f.: exp(x)*x*(BesselI(0, 2*x) + BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012
a(n) = 2*n/(n-1)*a(n-1) + 3*a(n-2) for n>=2, a(n) = n for n<2. a(n) = n*A005773(n). - Alois P. Heinz, Jul 15 2013
a(n) ~ 3^(n-1/2)*sqrt(n/Pi). - Vaclav Kotesovec, Oct 08 2013
a(n) = (-1)^(n+1)*JacobiP(n-1,1,-n+1/2,-7). - Peter Luschny, Sep 23 2014

A107231 a(n) = C(n+2,2)*C(n,floor(n/2)).

Original entry on oeis.org

1, 3, 12, 30, 90, 210, 560, 1260, 3150, 6930, 16632, 36036, 84084, 180180, 411840, 875160, 1969110, 4157010, 9237800, 19399380, 42678636, 89237148, 194699232, 405623400, 878850700, 1825305300, 3931426800, 8143669800, 17450721000

Offset: 0

Paul Barry, May 13 2005

Third column of A107230. Related to the generalized pentagonal numbers A001318. The sequence 0,0,1,3,12,... is an inverse Chebyshev transform of 0,0,1,3,8,... (see A034828). This transform maps a g.f. g(x) to (1/sqrt(1-4x^2))g(c(x^2)). Thus A001318, as first differences of A034828, can be expressed in terms of A107231.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000217, A001405.

Cf. A001318, A034828, A107230.

Table[Binomial[n + 2, 2]*Binomial[n, Floor[n/2]], {n,0,50}] (* G. C. Greubel, Jun 13 2017 *)

for(n=0,50, print1(binomial(n+2,2)*binomial(n,n\2), ", ")) \\ G. C. Greubel, Jun 13 2017

G.f.: (1+x)*(1-sqrt(1-4*x^2))^3*(sqrt(1-4*x^2)-4*x^2+1)^2/(8*x^4*(1-4*x^2)^(5/2)*(sqrt(1-4*x^2)+2*x-1)^2).
a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n+2, k)*A034828(n+2-2*k). [corrected by Jason Yuen, Sep 02 2024]
Conjecture: n*a(n) +(n-4)*a(n-1) +2*(-2*n-5)*a(n-2) -4*n*a(n-3)=0. - R. J. Mathar, Nov 24 2012
G.f.: (1+x)/((1+2*x)^(3/2)*(1-2*x)^(5/2)). - Vladimir Reshetnikov, Aug 01 2018
Sum_{n>=0} 1/a(n) = Pi^2/9 - 2*Pi/sqrt(3) + 4. - Amiram Eldar, Sep 03 2024

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1

Offset: 0

Philippe Deléham, Dec 19 2009

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Cf. A107230, A171651

Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

T[n_,k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0,2*x]+BesselI[1,2*x])*x^k / k!,{x,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.
Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013
E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023

cp's OEIS Frontend

A107238 A Chebyshev transform of number triangle A107230.

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A132894 Number of (1,0) steps in all paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., in all length-n left factors of Motzkin paths).

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A107231 a(n) = C(n+2,2)*C(n,floor(n/2)).

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A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

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