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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.

%I A257607 #19 Mar 25 2022 02:18:18
%S A257607 1,5,5,25,60,25,125,535,535,125,625,4210,7490,4210,625,3125,30885,
%T A257607 86110,86110,30885,3125,15625,216560,880735,1377760,880735,216560,
%U A257607 15625,78125,1471235,8330745,18948695,18948695,8330745,1471235,78125,390625,9764910,74498800,234897010,341076510,234897010,74498800,9764910,390625
%N 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.
%H A257607 G. C. Greubel, <a href="/A257607/b257607.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257607 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.
%F A257607 Sum_{k=0..n} T(n, k) = A049198(n).
%F A257607 From _G. C. Greubel_, Mar 24 2022: (Start)
%F A257607 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.
%F A257607 T(n, n-k) = T(n, k).
%F A257607 T(n, 0) = A000351(n).
%F A257607 T(n, 1) = 10*6^n - 5^n*(10 + n).
%F A257607 T(n, 2) = 55*7^n - 10*6^n*(n+10) + 5^n*binomial(n+10, 2). (End)
%e A257607 Triangle begins as:
%e A257607       1;
%e A257607       5,       5;
%e A257607      25,      60,      25;
%e A257607     125,     535,     535,      125;
%e A257607     625,    4210,    7490,     4210,      625;
%e A257607    3125,   30885,   86110,    86110,    30885,    3125;
%e A257607   15625,  216560,  880735,  1377760,   880735,  216560,   15625;
%e A257607   78125, 1471235, 8330745, 18948695, 18948695, 8330745, 1471235, 78125;
%t A257607 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]]];
%t A257607 Table[T[n,k,1,5], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 24 2022 *)
%o A257607 (Sage)
%o A257607 def T(n,k,a,b): # A257607
%o A257607     if (k<0 or k>n): return 0
%o A257607     elif (n==0): return 1
%o A257607     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257607 flatten([[T(n,k,1,5) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 24 2022
%Y A257607 Cf. A000351, A008292, A049198 (row sums), A256890, A257180, A257606
%Y A257607 Cf. A257615, A257624
%Y A257607 Similar sequences listed in A256890.
%K A257607 nonn,tabl
%O A257607 0,2
%A A257607 _Dale Gerdemann_, May 03 2015