A101124 Number triangle associated to Chebyshev polynomials of first kind.
1, 0, 1, -1, 1, 1, 0, 1, 2, 1, 1, 1, 7, 3, 1, 0, 1, 26, 17, 4, 1, -1, 1, 97, 99, 31, 5, 1, 0, 1, 362, 577, 244, 49, 6, 1, 1, 1, 1351, 3363, 1921, 485, 71, 7, 1, 0, 1, 5042, 19601, 15124, 4801, 846, 97, 8, 1, -1, 1, 18817, 114243, 119071, 47525, 10081, 1351, 127, 9, 1, 0, 1, 70226, 665857, 937444, 470449, 120126, 18817, 2024, 161
Offset: 0
Examples
As a number triangle, rows begin:
{1},
{0,1},
{-1,1,1},
{0,1,2,1},
...
As a square array, rows begin
1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, ...
-1, 1, 7, 17, 31, ...
0, 1, 26, 99, 244, ...
1, 1, 97, 577, 1921, ...
Links
Crossrefs
Programs
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Mathematica
T[n_, k_] := SeriesCoefficient[x^k (1 - k x)/(1 - 2 k x + x^2), {x, 0, n}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 12 2017 *)
Formula
Number triangle S(n, k)=T(n-k, k), k
Columns have g.f. x^k(1-kx)/(1-2kx+x^2).
Also, square array if(n=0, 1, T(n, k)) read by antidiagonals.
A323613 Antidiagonal sums of A323182.
1, 1, 2, 8, 27, 105, 492, 2584, 14893, 93625, 637342, 4663856, 36455959, 302825585, 2661650680, 24662914640, 240141823417, 2450053360913, 26125165902810, 290487741343352, 3361177509359859, 40396577112990745, 503447944487902244, 6496090993661295784, 86660903426459105701
Offset: 0
Keywords
Examples
a(1) = 0 + 1 = 1. a(2) = -1 + 2 + 1 = 2. a(3) = 0 + 3 + 4 + 1 = 8.
Links
Programs
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Mathematica
Table[Sum[ChebyshevU[k, n - k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jan 20 2019 *) -
PARI
{a(n) = sum(k=0, n, polchebyshev(k, 2, n-k))}
A341576 a(n) = Sum_{k=0..n} U_k((n-k)/2) where U_n(x) is a Chebyshev polynomial of the 2nd kind.
1, 1, 1, 3, 7, 16, 46, 149, 520, 1977, 8136, 35878, 168501, 838945, 4409957, 24385913, 141412615, 857611640, 5426144190, 35739397739, 244573978100, 1735854397529, 12757309001220, 96941738970956, 760649367654461, 6155205917196409, 51308394497243469
Offset: 0
Keywords
Programs
-
Mathematica
a[n_] := Sum[ChebyshevU[k, (n - k)/2], {k, 0, n}]; Array[a, 27, 0] (* Amiram Eldar, Mar 08 2021 *) -
PARI
a(n) = sum(k=0, n, polchebyshev(k, 2, (n-k)/2));
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Python
from fractions import Fraction from sympy import chebyshevu def A341576(n): return sum(chebyshevu(k,Fraction(n-k,2)) for k in range(n+1)) # Chai Wah Wu, Nov 08 2023