A034839 Triangular array formed by taking every other term of each row of Pascal's triangle.
1, 1, 1, 1, 1, 3, 1, 6, 1, 1, 10, 5, 1, 15, 15, 1, 1, 21, 35, 7, 1, 28, 70, 28, 1, 1, 36, 126, 84, 9, 1, 45, 210, 210, 45, 1, 1, 55, 330, 462, 165, 11, 1, 66, 495, 924, 495, 66, 1, 1, 78, 715, 1716, 1287, 286, 13
Offset: 0
Examples
Triangular array begins: 1 1 1 1 1 3 1 6 1 1 10 5 1 15 15 1 ... cosh(4x) = (cosh x)^5 + 10 (cosh x)^3 (sinh x)^2 + 5 (cosh x) (sinh x)^4, so row 4 is (1,10,5). See Mathematica program. - _Clark Kimberling_, Aug 03 2024
Links
- G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
- M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
- H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
- Tom Copeland, Juggling Zeros in the Matrix: Example II, 2020.
- C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.
- Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.
- S.-M. Ma, On some binomial coefficients related to the evaluation of tan(nx), arXiv preprint arXiv:1205.0735 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 13 2012
- K. Oliver and H. Prodinger, The continued fraction expansion of Gauss' hypergeometric function and a new application to the tangent function, Transactions of the Royal Society of South Africa, Vol. 76 (2012), 151-154, [DOI]; [PDF]. - From _N. J. A. Sloane_, Jan 03 2013
- Eric Weisstein's World of Mathematics, Tangent [From _Eric W. Weisstein_, Oct 18 2008]
Crossrefs
Programs
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Magma
/* As a triangle */ [[Binomial(n,2*k):k in [0..Floor(n/2)]] : n in [0..10]]; // G. C. Greubel, Feb 23 2018
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Maple
for n from 0 to 13 do seq(binomial(n,2*k),k=0..floor(n/2)) od;# yields sequence in triangular form; # Emeric Deutsch, Mar 30 2005
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 12; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x] v[n_, x_] := u[n - 1, x] + v[n - 1, x] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] (* A034839 as a triangle *) cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] (* A034867 as a triangle *) (* Clark Kimberling, Feb 18 2012 *) Table[Binomial[n, k], {n, 0, 13}, {k, 0, Floor[n, 2], 2}] // Flatten (* Michael De Vlieger, Dec 13 2016 *) (* The triangle gives coefficients for cosh(nx) as a linear combination of products (cosh(x)^h)*(sinh(x)^k) *) Column[Table[TrigExpand[Cosh[n x]], {n, 0, 10}]] (* Clark Kimberling, Aug 03 2024 *) -
PARI
for(n=0,15, for(k=0,floor(n/2), print1(binomial(n, 2*k), ", "))) \\ G. C. Greubel, Feb 23 2018
Formula
E.g.f.: exp(x)*cosh(x*sqrt(y)). - Vladeta Jovovic, Mar 20 2005
From Emeric Deutsch, Mar 30 2005: (Start)
T(n, k) = binomial(n, 2*k), for n >= 0 and k = 0, 1, ..., floor(n/2).
G.f.: (1-z)/((1-z)^2 - t*z^2). (End)
O.g.f. for column no. k is (1/(1-x))*(x/(1-x))^(2*k), k >= 0 [from the g.f. given in the preceding formula]. - Wolfdieter Lang, Jan 18 2013
From Peter Bala, Jul 14 2015: (Start)
Stretched Riordan array ( 1/(1 - x ), x^2/(1 - x)^2 ) in the terminology of Corsani et al.
P * transpose(P) is A119326 read as a square array.
Let Q denote the array ( (-1)^k*binomial(2*n,k) )n,k>=0. Q is a signed version of A034870. Then Q*P = the identity matrix, that is, Q is a left-inverse array of P (see Corsani et al., p. 111).
Even rows are A086645. An aerated version of this array is A099174 with each diagonal divided by the first element of the diagonal, the double factorials A001147. - Tom Copeland, Dec 12 2015
Comments