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%I #71 May 04 2022 21:12:30
%S 1,1,3,1,10,5,1,21,35,7,1,36,126,84,9,1,55,330,462,165,11,1,78,715,
%T 1716,1287,286,13,1,105,1365,5005,6435,3003,455,15,1,136,2380,12376,
%U 24310,19448,6188,680,17,1,171,3876,27132,75582,92378,50388,11628,969,19,1,210,5985,54264,203490,352716,293930,116280,20349,1330,21
%N Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
%C The row polynomials Pe(n, x) := Sum_{m=0..n} a(n, m)*x^m appear as numerators of the generating functions for the even numbered column sequences of array A034870.
%C Elements have the same parity as those of Pascal's triangle.
%C All zeros of polynomial Pe(n, x) are negative. They are -tan^2(Pi/2*n+1), -tan^2(2*Pi/2*n+1), ..., -tan^2(n*Pi/2*n+1). Moreover, for m >= 1, Pe(m, -x^2) is the characteristic polynomial of the linear difference equation with constant coefficients for differences between multiples of 2*m+1 with even and odd digit sum in base 2*m in the interval [0,(2*m)^n). - _Vladimir Shevelev_ and _Peter J. C. Moses_, May 22 2012
%C Row reverse of A103327. - _Peter Bala_, Jul 29 2013
%C The row polynomial Pe(d, x), multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,1] given in A048854. - _Wolfdieter Lang_, Oct 12 2017
%D A. M. Yaglom and I. M. Yaglom, An elementary proof of the Wallis, Leibniz and Euler formulas for pi. Uspekhi Matem. Nauk, VIII (1953), 181-187(in Russian).
%H Indranil Ghosh, <a href="/A091042/b091042.txt">Rows 0..120 of triangle, flattened</a>
%H Wolfdieter Lang, <a href="/A091042/a091042.txt">First 9 rows</a>.
%H V. Shevelev, <a href="http://arxiv.org/abs/0710.3177">On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m</a>, arXiv:0710.3177 [math.NT], 2007.
%H V. Shevelev and P. Moses, <a href="http://arxiv.org/abs/1207.0404">Tangent power sums and their applications</a>, arXiv:1207.0404 [math.NT], 2012-2014.
%F T(n, m) = binomial(2*n+1, 2*m) = A007318(2*n+1, 2*m), n >= m >= 0, otherwise 0.
%F From _Peter Bala_, Jul 29 2013: (Start)
%F E.g.f.: sinh(t)*cosh(sqrt(x)*t) = t + (1 + 3*x)*t^3/3! + (1 + 10*x + 5*x^2)*t^5/5! + (1 + 21*x + 35*x^2 + 7*x^3)*t^7/7! + ....
%F O.g.f.: A(x,t) = (1 + (x - 1)*t)/( (1 + (x - 1)*t)^2 - 4*t*x ) = 1 + (1 + 3*x)*t + (1 + 10*x + 5*x^2)*t^2 + ...
%F The function A( x/(x + 4), t*(x + 4)/4 ) = 1 + (1 + x)*t + (1 + 3*x + x^2)*t^2 + ... is the o.g.f. for A085478.
%F O.g.f. for n-th diagonal: ( Sum_{k = 0..n} binomial(2*n,2*k)*x^k )/(1 - x)^(2*n).
%F n-th row polynomial R(n,x) = (1/2)*( (1 + sqrt(x))^(2*n+1) - (sqrt(x) - 1)^(2*n+1) ).
%F Row sums A000302. (End)
%F T(n, k) = 2*T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-1) - T(n-2,k) - T(n-2,k-2) with T(0,0)=T(1,0)=1, T(1,1)=3, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Nov 26 2013
%F From _Peter Bala_, Jan 31 2022: (Start)
%F Define S(r,N) = Sum_{j = 1..N} j^r. Then the following identity holds for n >= 0: (1/2)*(N^2 + N)^(2*n+1) = T(n,0)*S(2*n+1,N) + T(n,1)*S(2*n+3,N) + ... + T(n,n)* S(4*n+1,N). Some examples are given below. (End)
%e Triangle a(n, m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: 1 3
%e 2: 1 10 5
%e 3: 1 21 35 7
%e 4: 1 36 126 84 9
%e 5: 1 55 330 462 165 11
%e 6: 1 78 715 1716 1287 286 13
%e 7: 1 105 1365 5005 6435 3003 455 15
%e 8: 1 136 2380 12376 24310 19448 6188 680 17
%e 9: 1 171 3876 27132 75582 92378 50388 11628 969 19
%e 10: 1 210 5985 54264 203490 352716 293930 116280 20349 1330 21
%e ... reformatted and extended. - _Wolfdieter Lang_, Oct 12 2017
%e From _Peter Bala_, Jan 30 2022: (Start)
%e (1/2)*(N^2 + N) = Sum_{j = 1..N} j.
%e (1/2)*(N^2 + N)^3 = Sum_{j = 1..N} j^3 + 3*Sum_{j = 1..N} j^5.
%e (1/2)*(N^2 + N)^5 = Sum_{j = 1..N} j^5 + 10*Sum_{j = 1..N} j^7 + 5*Sum_{j = 1..N} j^9.
%e (1/2)*(N^2 + N)^7 = Sum_{j = 1..N} j^7 + 21*Sum_{j = 1..N} j^9 + 35*Sum_{j = 1..N} j^11 + 7*Sum_{j = 1..N} j^13. (End)
%p f := (x, t) -> cosh(sqrt(x)*t)*sinh(t); seq(seq(coeff(((2*n+1)!*coeff(series(f(x,t),t,2*n+2),t,2*n+1)),x,k),k=0..n),n=0..9); # _Peter Luschny_, Jul 29 2013
%t T[n_, k_] /; 0 <= k <= n := T[n, k] = 2T[n-1, k] + 2T[n-1, k-1] + 2T[n-2, k-1] - T[n-2, k] - T[n-2, k-2]; T[0, 0] = T[1, 0] = 1; T[1, 1] = 3; T[_, _] = 0;
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _Jean-François Alcover_, Jul 29 2018, after _Philippe Deléham_ *)
%t Table[Binomial[2*n+1, 2*k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 01 2019 *)
%o (PARI) T(n,k) = binomial(2*n+1, 2*k); \\ _G. C. Greubel_, Aug 01 2019
%o (Magma) [[Binomial(2*n+1,2*k): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) [[binomial(2*n+1, 2*k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Aug 01 2019
%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k) ))); # _G. C. Greubel_, Aug 01 2019
%Y Cf. A212500, A038754. A000302 (row sums), A085478, A103327 (row reverse), A048854, A103328.
%K nonn,easy,tabl
%O 0,3
%A _Wolfdieter Lang_, Jan 23 2004
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