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A119467
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A masked Pascal triangle.
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7
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1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 5, 0, 10, 0, 1, 1, 0, 15, 0, 15, 0, 1, 0, 7, 0, 35, 0, 21, 0, 1, 1, 0, 28, 0, 70, 0, 28, 0, 1, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 1, 0, 45, 0, 210, 0, 210, 0, 45, 0, 1, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 1, 0, 66, 0, 495, 0, 924
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Row sums are A011782. Diagonal sums are F(n+1)*(1+(-1)^n)/2 (aerated version of A001519). Product by Pascal's triangle A007318 is A119468. Schur product of (1/(1-x),x/(1-x)) and (1/(1-x^2),x).
Exponential Riordan array (cosh(x),x). Inverse is (sech(x),x) or A119879. - Paul Barry (pbarry(AT)wit.ie), May 26 2006
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: Rows give coefficients of polynomials p_n(x) = sum_{k=0..n} (k+1 mod 2)*binomial(n,k)*x^(n-k)having e.g.f. exp(x*t)*cosh(t) = 1*(t^0/0!)+x*(t^1/1!)+(1+x^2)*(t^2/2!)+...
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FORMULA
| G.f.: (1-xy)/(1-2xy-x^2+x^2*y^2); T(n,k)=C(n,k)*(1+(-1)^(n-k))/2; Column k has g.f. (1/(1-x^2)(x/(1-x^2))^k*sum{j=0..k+1, C(k+1,j)*sin((j+1)*pi/ 2)^2*x^j};
Column k has e.g.f. cosh(x)*x^k/k! - Paul Barry (pbarry(AT)wit.ie), May 26 2006
Let Pascal's triangle, A007318 = P; then this triangle = (1/2) * (P + 1/P). Also A131047 = (1/2) * (P - 1/P). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2007
Equals A007318 - A131047 since the zeros of the triangle are masks for the terms of A131047. Thus A119467 + A131047 = Pascal's triangle. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2007
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EXAMPLE
| Triangle begins
1,
0, 1,
1, 0, 1,
0, 3, 0, 1,
1, 0, 6, 0, 1,
0, 5, 0, 10, 0, 1,
1, 0, 15, 0, 15, 0, 1,
0, 7, 0, 35, 0, 21, 0, 1,
1, 0, 28, 0, 70, 0, 28, 0, 1,
0, 9, 0, 84, 0, 126, 0, 36, 0, 1,
1, 0, 45, 0, 210, 0, 210, 0, 45, 0, 1
p[0](x) = 1
p[1](x) = x
p[2](x) = 1+x^2
p[3](x) = 3*x+x^3
p[4](x) = 1+6*x^2+x^4
p[5](x) = 5*x+10*x^3+x^5
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MAPLE
| Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)
# Polynomials: p_n(x)
p := proc(n, x) local k, pow;
pow := (n, k) -> `if`(n=0 and k=0, 1, n^k);
add((k+1 mod 2)*binomial(n, k)*pow(x, n-k), k=0..n) end;
# Coefficients: a(n)
seq(print(seq(coeff(i!*coeff(series(exp(x*t)*cosh(t), t, 16), t, i), x, n), n=0..i)), i=0..8); (End)
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CROSSREFS
| Cf. A131047.
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)
Cf. A034839, A162590.
p[n](k), n=0,1,...
k=0: 1,0,1,0,1,0,......... A128174
k=1: 1,1,2,4,8,16,........ A011782
k=2: 1,2,5,14,41,122,..... A007051
k=3: 1,3,10,36,136,....... A007582
k=4: 1,4,17,76,353,....... A081186
k=5: 1,5,26,140,776,...... A081187
k=6: 1,6,37,234,1513,..... A081188
k=7: 1,7,50,364,2696,..... A081189
k=8: 1,8,65,536,4481,..... A081190
k=9: 1,9,82,756,7048,..... A060531
k=10:1,10,101,1030,....... A081192
p[n](k), k=0,1,...
p[0]: 1,1,1,1,1,1, ....... A000012
p[1]: 0,1,2,3,4,5, ....... A001477
p[2]: 1,2,5,10,17,26, .... A002522
p[3]: 0,4,14,36,76,140, .. A079908 (End)
Sequence in context: A094675 A200472 A112743 * A166353 A110235 A036856
Adjacent sequences: A119464 A119465 A119466 * A119468 A119469 A119470
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), May 21 2006
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EXTENSIONS
| Edited by N. J. A. Sloane, Jul 14 2009
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