login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A257607
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 + 5.
6
1, 5, 5, 25, 60, 25, 125, 535, 535, 125, 625, 4210, 7490, 4210, 625, 3125, 30885, 86110, 86110, 30885, 3125, 15625, 216560, 880735, 1377760, 880735, 216560, 15625, 78125, 1471235, 8330745, 18948695, 18948695, 8330745, 1471235, 78125, 390625, 9764910, 74498800, 234897010, 341076510, 234897010, 74498800, 9764910, 390625
OFFSET
0,2
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 + 5.
Sum_{k=0..n} T(n, k) = A049198(n).
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 = 5.
T(n, n-k) = T(n, k).
T(n, 0) = A000351(n).
T(n, 1) = 10*6^n - 5^n*(10 + n).
T(n, 2) = 55*7^n - 10*6^n*(n+10) + 5^n*binomial(n+10, 2). (End)
EXAMPLE
Triangle begins as:
1;
5, 5;
25, 60, 25;
125, 535, 535, 125;
625, 4210, 7490, 4210, 625;
3125, 30885, 86110, 86110, 30885, 3125;
15625, 216560, 880735, 1377760, 880735, 216560, 15625;
78125, 1471235, 8330745, 18948695, 18948695, 8330745, 1471235, 78125;
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, 5], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
PROG
(Sage)
def T(n, k, a, b): # A257607
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, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022
CROSSREFS
Similar sequences listed in A256890.
Sequence in context: A237654 A038247 A263798 * A093643 A223263 A189318
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 03 2015
STATUS
approved