A257624
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) = 3*x + 5.
Original entry on oeis.org
1, 5, 5, 25, 80, 25, 125, 915, 915, 125, 625, 9070, 20130, 9070, 625, 3125, 83185, 348410, 348410, 83185, 3125, 15625, 727980, 5246655, 9755480, 5246655, 727980, 15625, 78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125
Offset: 0
Triangle begins as:
1;
5, 5;
25, 80, 25;
125, 915, 915, 125;
625, 9070, 20130, 9070, 625;
3125, 83185, 348410, 348410, 83185, 3125;
15625, 727980, 5246655, 9755480, 5246655, 727980, 15625;
78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125;
Similar sequences listed in
A256890.
-
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,3,5], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
-
def T(n,k,a,b): # A257624
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,3,5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
A257615
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) = 2*x + 5.
Original entry on oeis.org
1, 5, 5, 25, 70, 25, 125, 715, 715, 125, 625, 6380, 12870, 6380, 625, 3125, 52785, 186010, 186010, 52785, 3125, 15625, 416370, 2360295, 4092220, 2360295, 416370, 15625, 78125, 3180215, 27488205, 75698255, 75698255, 27488205, 3180215, 78125
Offset: 0
Triangle begins as:
1;
5, 5;
25, 70, 25;
125, 715, 715, 125;
625, 6380, 12870, 6380, 625;
3125, 52785, 186010, 186010, 52785, 3125;
15625, 416370, 2360295, 4092220, 2360295, 416370, 15625;
78125, 3180215, 27488205, 75698255, 75698255, 27488205, 3180215, 78125;
Similar sequences listed in
A256890.
-
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,2,5], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)
-
def T(n,k,a,b): # A257610
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,2,5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022
A257626
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) = 3*x + 6.
Original entry on oeis.org
1, 6, 6, 36, 108, 36, 216, 1404, 1404, 216, 1296, 15876, 33696, 15876, 1296, 7776, 166212, 642492, 642492, 166212, 7776, 46656, 1659204, 10701720, 19274760, 10701720, 1659204, 46656, 279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936
Offset: 0
Triangle begins as:
1;
6, 6;
36, 108, 36;
216, 1404, 1404, 216;
1296, 15876, 33696, 15876, 1296;
7776, 166212, 642492, 642492, 166212, 7776;
46656, 1659204, 10701720, 19274760, 10701720, 1659204, 46656;
279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936;
See similar sequences listed in
A256890.
-
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,3,6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
-
def T(n,k,a,b): # A257626
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,3,6) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022