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A225356 Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows. 4

%I #31 Mar 19 2022 03:26:25

%S 1,1,1,1,-22,1,1,-75,-75,1,1,-236,1446,-236,1,1,-721,9822,9822,-721,1,

%T 1,-2178,58479,-201244,58479,-2178,1,1,-6551,325061,-2160227,-2160227,

%U 325061,-6551,1,1,-19672,1736668,-19971304,49441990,-19971304,1736668,-19672,1

%N Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

%H G. C. Greubel, <a href="/A225356/b225356.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.

%e The triangle begins:

%e 1;

%e 1, 1;

%e 1, -22, 1;

%e 1, -75, -75, 1;

%e 1, -236, 1446, -236, 1;

%e 1, -721, 9822, 9822, -721, 1;

%e 1, -2178, 58479, -201244, 58479, -2178, 1;

%e 1, -6551, 325061, -2160227, -2160227, 325061, -6551, 1;

%e 1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1;

%t (* First program *)

%t q[x_, n_]= (1-x)^(n+1)*Sum[(2*m+1)^n*x^m, {m, 0, Infinity}];

%t t[n_, m_]:= t[n, m]= Table[CoefficientList[q[x, k], x], {k,0,15}][[n+1, m+1]];

%t p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i <= Floor[n/2], (-1)^i*t[n, i], (-1)^(n-i+1)*t[n, i]]], {i,0,n}]/(1-x);

%t Flatten[Table[CoefficientList[p[x, n], x], {n,10}]]

%t (* Second Program *)

%t A060187[n_, k_]:= Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i,k}];

%t T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] +(-1)^k*A060187[n+2, k+1], T[n, n-k] ]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 18 2022 *)

%o (Sage)

%o def A060187(n,k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) )

%o @CachedFunction

%o def A225356(n,k):

%o if (k==0 or k==n): return 1

%o elif (k <= (n//2)): return A225356(n,k-1) + (-1)^k*A060187(n+2,k+1)

%o else: return A225356(n,n-k)

%o flatten([[A225356(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 18 2022

%Y Cf A007318, A060187, A159041.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, May 07 2013

%E Edited by _N. J. A. Sloane_, May 11 2013

%E Edited by _G. C. Greubel_, Mar 18 2022

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)