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A155491 Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows. 4
1, 1, 1, 1, 12, 1, 1, 78, 78, 1, 1, 415, 1820, 415, 1, 1, 2031, 27410, 27410, 2031, 1, 1, 9534, 330225, 959350, 330225, 9534, 1, 1, 43660, 3488884, 23935450, 23935450, 3488884, 43660, 1, 1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 3.
From G. C. Greubel, Apr 01 2022: (Start)
T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1).
T(n, n-k) = T(n, k). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 12, 1;
1, 78, 78, 1;
1, 415, 1820, 415, 1;
1, 2031, 27410, 27410, 2031, 1;
1, 9534, 330225, 959350, 330225, 9534, 1;
1, 43660, 3488884, 23935450, 23935450, 3488884, 43660, 1;
1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1;
MATHEMATICA
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]];
T[n_, k_, m_]:= Binomial[n+1, k]*t[n+1, k+1, m]/(k+1);
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 01 2022 *)
PROG
(Sage)
@CachedFunction
def t(n, k, m):
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)
def T(n, k, m): return binomial(n+1, k)*t(n+1, k+1, m)/(k+1)
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 01 2022
CROSSREFS
Cf. A001263 (m=0), A155467 (m=1), this sequence (m=3), A155493 (m=4).
Cf. A142458.
Sequence in context: A174672 A174151 A342890 * A142460 A156280 A166962
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 23 2009
EXTENSIONS
Edited by G. C. Greubel, Apr 01 2022
STATUS
approved

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Last modified May 25 10:45 EDT 2024. Contains 372788 sequences. (Running on oeis4.)