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 A127080 Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2,2k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1). 10
 1, 1, 1, 1, 1, -2, 1, 1, -1, -5, 1, 1, 0, -4, 12, 1, 1, 1, -3, 3, 43, 1, 1, 2, -2, -4, 28, -120, 1, 1, 3, -1, -9, 15, -15, -531, 1, 1, 4, 0, -12, 4, 48, -288, 1680, 1, 1, 5, 1, -13, -5, 75, -105, 105, 8601, 1, 1, 6, 2, -12, -12, 72, 24, -624, 3984, -30240, 1, 1, 7, 3, -9, -17, 45, 105, -735, 945, -945, -172965 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Comment from N. J. A. Sloane, Jan 29 2020: (Start) It looks like there was a missing 2 in the definition, which I have now corrected. The old definition was: (Wrong!) Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2, k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1). (Wrong!) (End) REFERENCES V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007. LINKS G. C. Greubel, Antidiagonals n = 0..100, flattened FORMULA E.g.f.: Sum_{k >= 0} Q(m,2k) x^k/k! = (1+4x)^((m-1)/2)/(1+2x)^(m/2), Sum_{k >= 0} Q(m,2k+1) x^k/k! = (1+4x)^((m-2)/2)/(1+2x)^((m+1)/2). EXAMPLE Array begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012) 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012) -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, ... (A023444) -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, ... (A023447) 12, 3, -4, -9, -12, -13, -12, -9, -4, 3, ... (A127146) 43, 28, 15, 4, -5, -12, -17, -20, -21, -20, ... (A127147) -120, -15, 48, 75, 72, 45, 0, -57, -120, -183, ... (A127148) -531, -288, -105, 24, 105, 144, 147, 120, 69, 0, ... 1680, 105, -624, -735, -432, 105, 720, 1281, 1680, 1833, ... MAPLE f:= proc(k) option remember; if `mod`(k, 2)=0 then k!/(k/2)! else 2^(k-1)*((k-1)/2)!*add(binomial(2*j, j)/8^j, j=0..((k-1)/2)) fi; end; Q:= proc(n, k) option remember; if n=0 then (-1)^binomial(k, 2)*f(k) elif k<2 then 1 elif `mod`(k, 2)=0 then (n-k+1)*Q(n+1, k-1) - (k-1)*Q(n+2, k-2) else ( (n-k+1)*Q(n+1, k-1) - (k-1)*(n+1)*Q(n+2, k-2) )/n fi; end; seq(seq(Q(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 30 2020 MATHEMATICA Q[0, k_]:= Q[0, k]= (-1)^Binomial[k, 2]*If[EvenQ[k], k!/(k/2)!, 2^(k-1)*((k-1)/2)!* Sum[Binomial[2*j, j]/8^j, {j, 0, (k-1)/2}] ]; Q[n_, k_]:= Q[n, k]= If[k<2, 1, If[EvenQ[k], (n-k+1)*Q[n+1, k-1] - (k-1)*Q[n+2, k-2], ((n -k+1)*Q[n+1, k-1] - (k-1)*(n+1)*Q[n+2, k-2])/n]]; Table[Q[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 30 2020 *) PROG (Sage) @CachedFunction def f(k): if (mod(k, 2)==0): return factorial(k)/factorial(k/2) else: return 2^(k-1)*factorial((k-1)/2)*sum(binomial(2*j, j)/8^j for j in (0..(k-1)/2)) def Q(n, k): if (n==0): return (-1)^binomial(k, 2)*f(k) elif (k<2): return 1 elif (mod(k, 2)==0): return (n-k+1)*Q(n+1, k-1) - (k-1)*Q(n+2, k-2) else: return ( (n-k+1)*Q(n+1, k-1) - (k-1)*(n+1)*Q(n+2, k-2) )/n [[Q(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 30 2020 CROSSREFS See A105937 for another version. Columns give A127137, A127138, A127144, A127145; Rows give A127146, A127147, A127148. Sequence in context: A330942 A141471 A331572 * A216645 A216635 A213945 Adjacent sequences: A127077 A127078 A127079 * A127081 A127082 A127083 KEYWORD sign,tabl AUTHOR N. J. A. Sloane, Mar 24 2007 EXTENSIONS More terms added by G. C. Greubel, Jan 30 2020 STATUS approved

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Last modified May 29 21:40 EDT 2023. Contains 363042 sequences. (Running on oeis4.)