A055829
a(n) = T(2n+5,n), array T as in A055818.
Original entry on oeis.org
1, 95, 768, 5216, 33024, 201792, 1208320, 7145792, 41919744, 244590496, 1421823232, 8243669664, 47708339712, 275738420864, 1592186658816, 9187634766976, 52992487665152, 305556178607328, 1761501729738496
Offset: 0
Cf.
A055818,
A055819,
A055820,
A055821,
A055822,
A055823,
A055824,
A055825,
A055826,
A055827,
A055828.
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T:= proc(i, j) option remember;
if i=0 or j=0 then 1
else add(add(T(h,m), m=0..j), h=0..i-1)
fi; end:
seq(T(n+5, n), n=0..30); # G. C. Greubel, Jan 23 2020
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T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n+5, n], {n,0,30}] (* G. C. Greubel, Jan 23 2020 *)
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@CachedFunction
def T(i, j):
if (i==0 or j==0): return 1
else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[T(n+5, n) for n in (0..30)] # G. C. Greubel, Jan 23 2020
A295288
Binomial transform of the centered triangular numbers A005448.
Original entry on oeis.org
1, 5, 19, 62, 184, 512, 1360, 3488, 8704, 21248, 50944, 120320, 280576, 647168, 1478656, 3350528, 7536640, 16842752, 37421056, 82706432, 181927936, 398458880, 869269504, 1889533952, 4093640704, 8841592832, 19042140160, 40902852608
Offset: 0
a(0) = (3*0^2 + 9*0 + 8)*2^(-3) = 8/8 = 1.
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.
- Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
- Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
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I:=[1,5,19]; [n le 3 select I[n] else 6*Self(n-1) -12*Self(n-2) +8*Self(n-3): n in [1..40]]; // G. C. Greubel, Oct 17 2018
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A:=n->(3*n^2+9*n+8)*2^(n-3); seq(A(n), n=0..70);
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Table[(3 n^2 + 9 n + 8) 2^(n-3), {n, 0, 70}]
LinearRecurrence[{6,-12,8}, {1,5,19}, 50] (* G. C. Greubel, Oct 17 2018 *)
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makelist((3*n^2 + 9*n + 8)*2^(n - 3), n, 0, 70);
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a(n) = (3*n^2 + 9*n + 8)*2^(n - 3) \\ Felix Fröhlich, Nov 19 2017
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