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A158454 Riordan array (1/(1-x^2), x/(1+x)^2). 15
1, 0, 1, 1, -2, 1, 0, 4, -4, 1, 1, -6, 11, -6, 1, 0, 9, -24, 22, -8, 1, 1, -12, 46, -62, 37, -10, 1, 0, 16, -80, 148, -128, 56, -12, 1, 1, -20, 130, -314, 367, -230, 79, -14, 1, 0, 25, -200, 610, -920, 771, -376, 106, -16, 1, 1, -30, 295, -1106, 2083, -2232, 1444, -574, 137, -18, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Coefficient table of the square of Chebyshev S-polynomials. For the S-polynomials see A049310, and for a proof see the array A181878, where the odd numbered rows are shifted by one to the left. - Wolfdieter Lang, Dec 15 2010
Image of the Catalan numbers A000108 by this matrix is the all 1's sequence.
Image of the central binomial numbers A000984 by this matrix is the counting numbers A000027.
Inverse array is the Riordan array (1-x^2*c(x)^4, xc(x)^2), where c(x) is the g.f. of A000108.
The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k belong to the class of Boas-Buck polynomials. Hence they satisfy the Boas-Buck identity: (E_x - n*1)*R(n, x) = -Sum_{p=0..n-1} ((1 - (-1)^p)*1 + 2*(-1)^(p+1)*E_x) R(n-1-p, x) for n >= 0. See the Boas-Buck comments and references in A046521. The ensuing recurrence for the column sequences is given in the formula section. - Wolfdieter Lang, Aug 10 2017
REFERENCES
Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
LINKS
Paul Barry and A. Hennessey, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
FORMULA
Number triangle T(n, k) = Sum_{j=0..n} (-1)^(j-k)*binomial(k+j, 2*k) = Sum_{j=0..n-k} (-1)^(n-k-j)*binomial(n+k-j, 2*k).
O.g.f. column k with leading zeros (Riordan array, see NAME): (1/(1-x^2))*(x/(1+x)^2)^k, k >= 0. - Wolfdieter Lang, Dec 15 2010
T(n, k) = (-1)^(n-k)*Sum_{j=0..floor(n/2)} binomial(n+k-1-2*j, 2*k-1), 0 <= k <= n, else 0. From the o.g.f. for column k after convolution. - Wolfdieter Lang, Dec 17 2010
O.g.f. row polynomials (rising powers in y):
((1+x)/(1-x))/(1+(2-y)*x+x^2) = Sum_{n>=0} (S(n,sqrt(y))^2*x^n, with Chebyshev S-polynomials from A049310. - Wolfdieter Lang, Dec 15 2010
Recurrences from the A- and Z-sequences for Riordan arrays. See the W. Lang link under A006232 for details and references.
T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), n >= 1.
T(n, k) = Sum_{j=0..n-k} A(j)*T(n-1, k-1+j), n >= k >= 1.
Here Z(0)=0 and Z(j) = A000108(j), j >= 1, (o.g.f. -1 + c(x), with the Catalan o.g.f. c(x)), and A(j) = A115141(j) = [1,-2,-1,-2,-5,-14,...], j >= 0, with o.g.f. 1/c(x)^2. - Wolfdieter Lang, Dec 20 2010
T(n, k) = Sum_{m=0..n} A129818(m, k), 0 <= k <= n. - Wolfdieter Lang, Dec 15 2010
Boas-Buck recurrence for column k: R(n, k) = (1/(n-k))*Sum_{p=k..n-1}((-1)^(n-p)*(2*k+1) + 1) * R(p, k), for n > k >= 0, with input R(k, k) = 1. See a comment above. - Wolfdieter Lang, Aug 10 2017
G.f.: (1 + x)/((1 - x)*(1 + x)^2 - t*x*(1 - x)). - G. C. Greubel, Dec 15 2018
T(n, k) = (-1)^(n - k)*binomial(k + n - 1, 2*k-1)*hypergeom([1, (k - n)/2, (1 + k - n)/2], [(1 - k - n)/2, (2 - k - n)/2], 1) for k >= 1 . - Peter Luschny, Aug 20 2022
EXAMPLE
The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10...
0: 1
1: 0 1
2: 1 -2 1
3: 0 4 -4 1
4: 1 -6 11 -6 1
5: 0 9 -24 22 -8 1
6: 1 -12 46 -62 37 -10 1
7: 0 16 -80 148 -128 56 -12 1
8: 1 -20 130 -314 367 -230 79 -14 1
9: 0 25 -200 610 -920 771 -376 106 -16 1
10: 1 -30 295 -1106 2083 -2232 1444 -574 137 -18 1
... Reformatted and extended by Wolfdieter Lang, Nov 24 2012
Recurrences (from A- and Z-sequences):
1 = T(6,0) = 0*0 + 1*9 +2*(-24) + 5*22 + 14*(-8)+ 42*1.
-80 = T(7,2) = 1*(-12) -2*(46) -1*(-62) -2*37 -5*(-10) -14*1. - Wolfdieter Lang, Dec 20 2010
MAPLE
A158454 := proc(n, k) (-1)^(n+k)*add(binomial(n+k-1-2*j, 2*k-1), j=0..floor(n/2)) ; end proc;
seq(seq(A158454(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Dec 17 2010
MATHEMATICA
nmax = 10; t[n_, k_] := (-1)^(n+k)* Sum[Binomial[n+k-1-2*j, 2*k-1], {j, 0, Floor[n/2]}]; t[n_?EvenQ, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 08 2011, after Maple *)
With[{m = 15}, CoefficientList[CoefficientList[Series[(1+x)/((1-x)*(1 + x)^2 -t*x*(1-x)), {x, 0, m}, {t, 0, m}], x], t]]//Flatten (* G. C. Greubel, Dec 15 2018 *)
T[n_, 0] := Boole[EvenQ[n]]; T[n_, k_] := (-1)^(n - k) Binomial[k+n-1, 2*k-1] HypergeometricPFQ[{1, (k - n)/2, (1 + k - n)/2}, {(1 - k - n)/2, (2 - k - n)/2}, 1]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // TableForm (* Peter Luschny, Aug 20 2022 *)
PROG
(PARI) {T(n, k) = sum(j=0, n, (-1)^(j-k)*binomial(k+j, 2*k))};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 15 2018
(Magma) [[(&+[(-1)^(j-k)*Binomial(k+j, 2*k): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Dec 15 2018
(Sage) [[sum((-1)^(j-k)*binomial(k+j, 2*k) for j in range(n+1)) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Dec 15 2018
(GAP) T:=Flat(List([0..10], n->List([0..n], k->Sum([0..n], j-> (-1)^(j-k)*Binomial(k+j, 2*k))))); # G. C. Greubel, Dec 15 2018
CROSSREFS
From Wolfdieter Lang, Aug 10 2017: (Start)
Row sums A011655(n+1), alternating row sums A007598(n+1)*(-1)^(n+1).
Column sequences k=0..5: A059841, A002620(n+2)*(-1)^(n), A001752(n)*(-1)^n, A001769(n)*(-1)^n, A001780(n)*(-1)^n, A001786(n)*(-1)^n. (End)
Sequence in context: A296129 A276544 A214753 * A049243 A077908 A052922
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Mar 19 2009
STATUS
approved

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Last modified April 21 14:03 EDT 2024. Contains 371870 sequences. (Running on oeis4.)