Original entry on oeis.org
1, 26, 2741, 683870, 315704418, 234725594388, 257237392999893, 390832857108454838, 787178784737043042806, 2031210797603911366282796, 6536955866068372922068141666, 25676217636579568989377656129516, 120915166829869713032692550819662756, 672580820552232143302651758669053327784
Offset: 1
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T[n_, k_, m_]:= T[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1,k-1,m] + (m*k-m+ 1)*T[n-1,k,m]];
a[n_]:= 2*T[2*n,n,3]/(n+1);
Table[a[n], {n,30}] (* modified by G. C. Greubel, Mar 14 2022 *)
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@CachedFunction
def T(n,k,m): # A142458
if (k==1 or k==n): return 1
else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
[2*T(2*n,n,3)/(n+1) for n in (1..30)] # G. C. Greubel, Mar 14 2022
Name corrected and more terms added by
G. C. Greubel, Mar 14 2022
A138076
Triangle read by rows: T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).
Original entry on oeis.org
1, -1, 1, 1, -6, 1, -1, 23, -23, 1, 1, -76, 230, -76, 1, -1, 237, -1682, 1682, -237, 1, 1, -722, 10543, -23548, 10543, -722, 1, -1, 2179, -60657, 259723, -259723, 60657, -2179, 1, 1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1, -1, 19673, -1756340, 21707972, -69413294, 69413294, -21707972, 1756340, -19673, 1
Offset: 0
Triangle begins as:
1;
-1, 1;
1, -6, 1;
-1, 23, -23, 1;
1, -76, 230, -76, 1;
-1, 237, -1682, 1682, -237, 1;
1, -722, 10543, -23548, 10543, -722, 1;
-1, 2179, -60657, 259723, -259723, 60657, -2179, 1;
1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1;
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A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
A138076:= func< n,k | (-1)^(n+k)*A060187(n+1,k+1) >;
[A138076(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2024
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p[t_] = Exp[t]*x/(Exp[2*t] + x);
Table[CoefficientList[(n!*(1+x)^(n+1)/x)*SeriesCoefficient[Series[p[ t], {t,0,30}], n], x], {n,0,12}]//Flatten
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@CachedFunction
def t(n,k): # t = A060187
if k==1 or k==n: return 1
return (2*(n-k)+1)*t(n-1, k-1) + (2*k-1)*t(n-1, k)
def A138076(n,k): return (-1)^(n+k)*t(n+1,k+1)
flatten([[A138076(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 21 2024
Original entry on oeis.org
1, -4, -40, 672, 8064, -253440, -3294720, 153753600, 2091048960, -130025226240, -1820353167360, 141707492720640, 2024392753152000, -189483161695027200, -2747505844577894400, 300609462994993152000, 4408938790593232896000
Offset: 0
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(* First program *)
p[x_, n_] = HermiteH[n, x] + ExpandAll[x^n*HermiteH[n, 1/x]];
b:= Table[CoefficientList[p[x, n], x], {n, 0, 50}];
Table[b[[2*n+2, n+1]]/(n+2), {n,0,20}]
(* Second program *)
A060821[n_, k_]:= If[EvenQ[n-k], (-1)^(Floor[(n-k)/2])*2^k*n!/(k!*(Floor[(n - k)/2]!)), 0];
a[n_]:= (A060821[2*n+1, n] + A060821[2*n+1, n+1])/(n+2);
Table[a[n], {n, 0, 25}] (* G. C. Greubel, Apr 04 2021 *)
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def A060821(n,k): return (-1)^((n-k)//2)*2^k*factorial(n)/(factorial(k)*factorial( (n-k)//2)) if (n-k)%2==0 else 0
def a(n): return (A060821(2*n+1, n) + A060821(2*n+1, n+1))/(n+2)
[a(n) for n in (0..25)] # G. C. Greubel, Apr 04 2021
Showing 1-3 of 3 results.
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