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 A103327 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1). 12
 1, 3, 1, 5, 10, 1, 7, 35, 21, 1, 9, 84, 126, 36, 1, 11, 165, 462, 330, 55, 1, 13, 286, 1287, 1716, 715, 78, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A subset of Pascal's triangle A007318. Elements have the same parity as those of Pascal's triangle. Matrix inverse is A104033. - Paul D. Hanna, Feb 28 2005 Row reverse of A091042. - Peter Bala, Jul 29 2013 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 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 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224. LINKS Indranil Ghosh, Rows 0..120 of triangle, flattened W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500. FORMULA 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 G.f.: A(x, y) = (1 + x*(1-y))/( (1 + x*(1-y))^2 - 4*x ). - Paul D. Hanna, Feb 28 2005 Sum_{k=0..n} T(n, k)*A000364(n-k) = A002084(n). - Philippe Deléham, Aug 27 2005 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 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 EXAMPLE The triangle T(n, k) begins: n\k   0    1     2      3      4      5      6     7    8   9  10 ... 0:    1 1:    3    1 2:    5   10     1 3:    7   35    21      1 4:    9   84   126     36      1 5:   11  165   462    330     55      1 6:   13  286  1287   1716    715     78      1 7:   15  455  3003   6435   5005   1365    105     1 8:   17  680  6188  19448  24310  12376   2380   136    1 9:   19  969 11628  50388  92378  75582  27132  3876  171   1 10:  21 1330 20349 116280 293930 352716 203490 54264 5985 210   1 ... reformatted and extended. - Wolfdieter Lang, Oct 12 2017 From Peter Bala, Aug 06 2013: (Start) Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n+1)! the column generating functions begin 1st col: cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + 9*y^4/9! + .... 2nd col: 1/3!*y*cosh(sqrt(y)) = y/3! + 10*y^2/5! + 35*y^3/7! + 84*y^4/9! + .... 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) MATHEMATICA Flatten[Table[Binomial[2n+1, 2k+1], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jun 19 2014 *) PROG (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 (PARI) T(n, k) = binomial(2*n+1, 2*k+1); for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 01 2019 (Maxima) create_list(binomial(2*n+1, 2*k+1), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */ (MAGMA) [Binomial(2*n+1, 2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019 (Sage) [[binomial(2*n+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019 CROSSREFS Reflected version of A091042. Cf. A086645, A103328. Cf. A104033, A086645, A292219. Sequence in context: A331432 A122366 A228781 * A177463 A065229 A233037 Adjacent sequences:  A103324 A103325 A103326 * A103328 A103329 A103330 KEYWORD nonn,easy,tabl AUTHOR Ralf Stephan, Feb 06 2005 STATUS approved

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Last modified August 14 17:36 EDT 2022. Contains 356122 sequences. (Running on oeis4.)