A010006 -id:A010006 - cp's OEIS frontend

A174169 A generalized Chebyshev transform of the Motzkin numbers A001006.

1, 1, -1, -2, 0, 0, -3, 1, 8, 1, 1, 26, 7, -51, -3, 0, -264, -186, 348, -120, -285, 2697, 2871, -2304, 3393, 8029, -25795, -36872, 16108, -60010, -159683, 213795, 413712, -181857, 833779, 2669534, -1272977, -4030235, 3611168, -9145271, -39467427

Offset: 0

Paul Barry, Mar 10 2010

Hankel transform is the (1,3) Somos-4 sequence A174170.

G.f.: (1-x+3x^2-sqrt(1-2x+3x^2-6x^3+9x^4))/(2x^2)=(1/(1+3x))*M(x/(1+3x^2)), M(x) the g.f. of A010006;
a(n) = sum{k=0..floor(n/2), (-3)^k*A001006(n-2k)}.
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +3*(n-1)*a(n-2) +3*(5-2*n)*a(n-3) +9*(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012

A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 4, 8, 25, 65, 197, 571, 1753, 5351, 16746, 52626, 167547, 536559, 1732272, 5622960, 18357211, 60205319, 198323708, 655787680, 2176141555, 7244106347, 24185285341, 80960692691, 271685400443, 913784117809, 3079889039230

Offset: 0

Paul Barry, Mar 10 2010

Hankel transform is the (1,8) Somos-4 sequence A097495(n+2).

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A001006.

Table[Sum[Binomial[n - k, k] 2^k * Hypergeometric2F1[(1 - #)/2, -#/2, 2, 4] &[n - 2 k], {k, 0, Floor[n/2]}], {n, 0, 26}] (* Michael De Vlieger, Feb 02 2017, after Peter Luschny at A001006 *)

G.f.: (1-x-2*x^2-sqrt(1-2*x-7*x^2+4*x^3+4*x^4))/(2*x^2) = (1/(1-2*x))*M(x/(1-2*x^2)), M(x) the g.f. of A010006.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * 2^k * A001006(n-2k).
Conjecture: (n+2)*a(n) -(2*n+1)*a(n-1) +7*(1-n)*a(n-2) +2*(2*n-5)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012
a(0) = a(1) = 1; a(n) = a(n-1) + 2 * a(n-2) + Sum_{k=0..n-2} a(k) * a(n-k-2). - Ilya Gutkovskiy, Nov 09 2021
a(n) ~ 17^(1/4) * (3 + sqrt(17))^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - Vaclav Kotesovec, Nov 11 2021

A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 18, 12, 2, 0, 1, 50, 66, 18, 2, 0, 1, 140, 330, 146, 24, 2, 0, 1, 392, 1610, 1070, 258, 30, 2, 0, 1, 1106, 7742, 7580, 2500, 402, 36, 2, 0, 1, 3138, 37058, 52556, 23330, 4850, 578, 42, 2, 0, 1, 8952, 177186, 360402, 212436, 56252, 8350, 786

Offset: 0

R. J. Mathar, Mar 02 2010

The values are computed starting with an auxiliary array which places the centered trinomial numbers A002426, the centered pentanomial numbers A005191, the centered 7-nomial numbers A025012 etc. into separate columns:
.1,....1,......1,.......1,........1,........1,.........1,.........1,.........1
.1,....3,......5,.......7,........9,.......11,........13,........15,........17
.1,....7,.....19,......37,.......61,.......91,.......127,.......169,.......217
.1,...19,.....85,.....231,......489,......891,......1469,......2255,......3281
.1,...51,....381,....1451,.....3951,.....8801,.....17151,.....30381,.....50101
.1,..141,...1751,....9331,....32661,....88913,....204763,....418503,....782153
.1,..393,...8135,...60691,...273127,...908755,...2473325,...5832765,..12354469
.1,.1107,..38165,..398567,..2306025,..9377467,..30162301,..82073295,.197018321
.1,.3139,.180325,.2636263,.19610233,.97464799,.370487485,1163205475,3164588407
This is a subarray of A077042. Rows are A005408, A003215, A063496, A083669 (see A077044) etc. The array A(k,n) is the first differences along each row of this auxiliary array.

			A(k,n) starts in row k=0, column n=0 as:
1,....0,......0,.......0,........0,........0,.........0,.........0,.........0
1,....2,......2,.......2,........2,........2,.........2,.........2,.........2
1,....6,.....12,......18,.......24,.......30,........36,........42,........48
1,...18,.....66,.....146,......258,......402,.......578,.......786,......1026
1,...50,....330,....1070,.....2500,.....4850,......8350,.....13230,.....19720
1,..140,...1610,....7580,....23330,....56252,....115850,....213740,....363650
1,..392,...7742,...52556,...212436,...635628,...1564570,...3359440,...6521704
1,.1106,..37058,..360402,..1907458,..7071442,..20784834,..51910994,.114945026
1,.3138,.177186,.2455938,.16973970,.77854566,.273022686,.792717990,2001382932

R. J. Mathar, Point counts of D_k and some A_k and E_K integer.., arXiv:1002.3844 [math .GT], variable A_k^s(n).

Cf. A008458 (row k=2), A010006 (row k=3), A110907.

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A174169 A generalized Chebyshev transform of the Motzkin numbers A001006.

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A174171 A generalized Chebyshev transform of the Motzkin numbers A001006.

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A175197 Array A(k,n) of the number of points of the A_k lattice with maximum infinity norm n, read by antidiagonals.

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