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%I #51 Sep 08 2022 08:45:16
%S 1,3,1,5,10,1,7,35,21,1,9,84,126,36,1,11,165,462,330,55,1,13,286,1287,
%T 1716,715,78,1,15,455,3003,6435,5005,1365,105,1,17,680,6188,19448,
%U 24310,12376,2380,136,1,19,969,11628,50388,92378,75582,27132,3876,171,1
%N Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
%C A subset of Pascal's triangle A007318.
%C Elements have the same parity as those of Pascal's triangle.
%C Matrix inverse is A104033. - _Paul D. Hanna_, Feb 28 2005
%C Row reverse of A091042. - _Peter Bala_, Jul 29 2013
%C Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. Cf. A086645. - _Peter Bala_, Aug 06 2013
%C The row polynomial P(d, x) = Sum_{k=0..d} T(d, k)*x^k, 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[4,3] given in A292219. - _Wolfdieter Lang_, Oct 12 2017
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
%H Indranil Ghosh, <a href="/A103327/b103327.txt">Rows 0..120 of triangle, flattened</a>
%H W. Wang and T. Wang, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
%F G.f. for column k: Sum_{j=0..k+1} C(2*(k+1), 2*j)x^j)/(1-x)^(2*(k+1)). - _Paul Barry_, Feb 24 2005
%F G.f.: A(x, y) = (1 + x*(1-y))/( (1 + x*(1-y))^2 - 4*x ). - _Paul D. Hanna_, Feb 28 2005
%F Sum_{k=0..n} T(n, k)*A000364(n-k) = A002084(n). - _Philippe Deléham_, Aug 27 2005
%F E.g.f.: 1/sqrt(x)*sinh(sqrt(x)*t)*cosh(t) = t + (3 + x)*t^3/3! + (5 + 10*x + x^2)*t^5/5! + .... - _Peter Bala_, Jul 29 2013
%F T(n+2,k+2) = 2*T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k+2) + 2*T(n,k+1) - T(n,k). - _Emanuele Munarini_, Jul 05 2017
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: 3 1
%e 2: 5 10 1
%e 3: 7 35 21 1
%e 4: 9 84 126 36 1
%e 5: 11 165 462 330 55 1
%e 6: 13 286 1287 1716 715 78 1
%e 7: 15 455 3003 6435 5005 1365 105 1
%e 8: 17 680 6188 19448 24310 12376 2380 136 1
%e 9: 19 969 11628 50388 92378 75582 27132 3876 171 1
%e 10: 21 1330 20349 116280 293930 352716 203490 54264 5985 210 1
%e ... reformatted and extended. - _Wolfdieter Lang_, Oct 12 2017
%e From _Peter Bala_, Aug 06 2013: (Start)
%e Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n+1)! the column generating functions begin
%e 1st col: cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + 9*y^4/9! + ....
%e 2nd col: 1/3!*y*cosh(sqrt(y)) = y/3! + 10*y^2/5! + 35*y^3/7! + 84*y^4/9! + ....
%e 3rd col: 1/5!*y^2*cosh(sqrt(y)) = y^2/5! + 21*y^3/7!! + 126*y^4/9! + 462*y^5/11! + .... (End)
%t Flatten[Table[Binomial[2n+1,2k+1],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Jun 19 2014 *)
%o (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1+X*(1-Y))/((1+X*(1-Y))^2-4*X),n,x),k,y)} \\ _Paul D. Hanna_, Feb 28 2005
%o (PARI) T(n,k) = binomial(2*n+1, 2*k+1);
%o for(n=0, 12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Aug 01 2019
%o (Maxima) create_list(binomial(2*n+1,2*k+1),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */
%o (Magma) [Binomial(2*n+1, 2*k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) [[binomial(2*n+1, 2*k+1) 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+1) ))); # _G. C. Greubel_, Aug 01 2019
%Y Reflected version of A091042. Cf. A086645, A103328.
%Y Cf. A104033, A086645, A292219.
%K nonn,easy,tabl
%O 0,2
%A _Ralf Stephan_, Feb 06 2005
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