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A372260
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Triangle read by rows: T(n, k) = (T(n-1, k-1) + T(n-1, k)) * 2 * n with initial values T(n, 0) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A001147(i) and T(i, j) = 0 if j > i.
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0
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1, 0, 2, 2, 8, 8, 8, 60, 96, 48, 60, 544, 1248, 1152, 384, 544, 6040, 17920, 24000, 15360, 3840, 6040, 79008, 287520, 503040, 472320, 230400, 46080, 79008, 1190672, 5131392, 11067840, 13655040, 9838080, 3870720, 645120, 1190672, 20314880, 101153024, 259187712, 395566080, 375889920, 219340800, 72253440, 10321920
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OFFSET
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0,3
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LINKS
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FORMULA
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T(n, 0) = (2*n - 2) * (T(n-1, 0 + T(n-2, 0)) for n > 1 with initial values T(n, 0) = 1 - n for n < 2 (see A053871).
T(n, k) = (Sum_{i=0..k} binomial(k, i) * T(n-i, 0)) * 2^(2*k) * binomial(n, k) / binomial(2*k, k).
E.g.f. of column k: (exp(-t) / sqrt(1 - 2*t)) * (2*t / (1 - 2*t))^k.
E.g.f.: exp((2*x / (1 - 2*t) - 1) * t) / sqrt(1 - 2*t).
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EXAMPLE
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Triangle T(n, k) starts:
n\k : 0 1 2 3 4 5 6 7
====================================================================
0 : 1
1 : 0 2
2 : 2 8 8
3 : 8 60 96 48
4 : 60 544 1248 1152 384
5 : 544 6040 17920 24000 15360 3840
6 : 6040 79008 287520 503040 472320 230400 46080
7 : 79008 1190672 5131392 11067840 13655040 9838080 3870720 645120
etc.
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MAPLE
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T := proc(n, k) option remember; `if`(k > n, 0, `if`(k = n, 2^n * n!, `if`(k = 0, `if`(n < 2, 1 - n, (2*n - 2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n))) end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 25 2024
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MATHEMATICA
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T[n_, k_]:=n!SeriesCoefficient[(Exp[-t]/Sqrt[1 - 2*t])*(2*t/(1-2*t))^k, {t, 0, n}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}]//Flatten (* Stefano Spezia, Apr 25 2024 *)
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PROG
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(PARI) { T(n, k) = if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1-n,
(2*n-2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n))) }
(PARI) memo = Map(); memoize(f, A[..]) =
{ my(res);
if(!mapisdefined(memo, [f, A], &res), res = call(f, A);
mapput(memo, [f, A], res)); res; }
T(n, k) =
{ if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1 - n,
(2 * n - 2) * (memoize(T, n-1, k) + memoize(T, n-2, k))),
(memoize(T, n-1, k-1) + memoize(T, n-1, k)) * 2 * n))); }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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