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A103327
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Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
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12
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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)
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
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LINKS
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Indranil Ghosh, Rows 0..120 of triangle, flattened
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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Flatten[Table[Binomial[2n+1, 2k+1], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jun 19 2014 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Ralf Stephan, Feb 06 2005
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STATUS
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approved
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