A257606 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) = x + 4.

1, 4, 4, 16, 40, 16, 64, 296, 296, 64, 256, 1928, 3552, 1928, 256, 1024, 11688, 34808, 34808, 11688, 1024, 4096, 67656, 302352, 487312, 302352, 67656, 4096, 16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384, 65536, 2076424, 18330496, 62617144, 93280000, 62617144, 18330496, 2076424, 65536

Offset

0,2

Links

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

Formula

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) = x + 4.

Sum_{k=0..n} T(n, k) = A049388(n).

T(n,0) = T(n,n) = 4^n. - Georg Fischer, Oct 02 2021

From G. C. Greubel, Mar 24 2022: (Start)

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 = 1, and b = 4.

T(n, n-k) = T(n, k).

T(n, 1) = 8*5^n - 4^n*(8+n).

T(n, 2) = 2*((56 +15*n +n^2)*4^(n-1) - 4*(8+n)*5^n + 3*6^(n+1)). (End)

Example

Triangle begins as:
      1;
      4,      4;
     16,     40,      16;
     64,    296,     296,      64;
    256,   1928,    3552,    1928,     256;
   1024,  11688,   34808,   34808,   11688,    1024;
   4096,  67656,  302352,  487312,  302352,   67656,   4096;
  16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384;

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, 1, 4], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

Prog

(Sage)
def T(n, k, a, b): # A257606
    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, 1, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Crossrefs

Cf. A008292, A049388 (row sums), A256890, A257180, A257607.

Cf. A257613, A257622.

Similar sequences listed in A256890.

Sequence in context: A099462 A218051 A092266 * A219398 A222104 A257613

Adjacent sequences: A257603 A257604 A257605 * A257607 A257608 A257609

Keyword

nonn,tabl

Author

Dale Gerdemann, May 03 2015

Extensions

a(3) corrected by Georg Fischer, Oct 02 2021

Status

approved