A257612 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.

1, 2, 2, 4, 24, 4, 8, 184, 184, 8, 16, 1216, 3680, 1216, 16, 32, 7584, 53824, 53824, 7584, 32, 64, 46208, 674752, 1507072, 674752, 46208, 64, 128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128, 256, 1677312, 84892672, 636233728, 1196803584, 636233728, 84892672, 1677312, 256

Offset

0,2

Comments

Corresponding entries in this triangle and in A060187 differ only by powers of 2. - F. Chapoton, Nov 04 2020

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.

Sum_{k=0..n} T(n,k) = A047053(n).

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 4, and b = 2. - G. C. Greubel, Mar 20 2022

Example

Triangle begins as:
    1;
    2,      2;
    4,     24,       4;
    8,    184,     184,        8;
   16,   1216,    3680,     1216,       16;
   32,   7584,   53824,    53824,     7584,      32;
   64,  46208,  674752,  1507072,   674752,   46208,     64;
  128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128;

Mathematica

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n, k, 4, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

Prog

(PARI) f(x) = 4*x + 2;
T(n, k) = t(n-k, k);
t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1, m) + f(n)*t(n, m-1)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); \\ Michel Marcus, May 06 2015
(Sage)
def T(n, k, a, b): # A257612
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)
flatten([[T(n, k, 4, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Crossrefs

Cf. A047053 (row sums), A060187, A142459, A257621.

Cf. A038208, A256890, A257609, A257610, A257614, A257616, A257617, A257618, A257619.

See similar sequences listed in A256890.

Sequence in context: A263439 A270527 A232275 * A009541 A307969 A212672

Adjacent sequences: A257609 A257610 A257611 * A257613 A257614 A257615

Keyword

nonn,tabl

Author

Dale Gerdemann, May 06 2015

Status

approved