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.
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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 *)
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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
A154693
Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).
Original entry on oeis.org
2, 3, 3, 5, 16, 5, 9, 66, 66, 9, 17, 260, 528, 260, 17, 33, 1026, 3624, 3624, 1026, 33, 65, 4080, 23820, 38656, 23820, 4080, 65, 129, 16302, 154548, 374856, 374856, 154548, 16302, 129, 257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260, 257
Offset: 0
The triangle begins as:
2;
3, 3;
5, 16, 5;
9, 66, 66, 9;
17, 260, 528, 260, 17;
33, 1026, 3624, 3624, 1026, 33;
65, 4080, 23820, 38656, 23820, 4080, 65;
129, 16302, 154548, 374856, 374856, 154548, 16302, 129;
257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260, 257;
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
- A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2501, (FIG. 3)
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A154693:= func< n,k | (2^(n-k) + 2^k)*EulerianNumber(n+1, k) >;
[A154693(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025
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p=2; q=1;
A008292[n_,k_]:= A008292[n,k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
T[n_, m_]:= (p^(n-m)*q^m + p^m*q^(n-m))*A008292[n+1,m+1];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten
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from sage.combinat.combinat import eulerian_number
def A154693(n,k): return (2^(n-k) +2^k)*eulerian_number(n+1,k)
print(flatten([[A154693(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 17 2025
Definition simplified by the Assoc. Eds. of the OEIS - Aug 08 2010.
A296229
Triangle read by rows: Eulerian triangle that produces sums of even powers.
Original entry on oeis.org
2, 4, 4, 8, 32, 8, 16, 176, 176, 16, 32, 832, 2112, 832, 32, 64, 3648, 19328, 19328, 3648, 64, 128, 15360, 152448, 309248, 152448, 15360, 128, 256, 63232, 1099008, 3998464, 3998464, 1099008, 63232, 256, 512, 257024, 7479296, 45175808, 79969280, 45175808, 7479296, 257024, 512, 1024, 1037312, 48988160
Offset: 1
The triangle T(n, k) begins:
n\k | 1 2 3 4 5 6 7 8
----+----------------------------------------------------
1 | 2
2 | 4 4
3 | 8 32 8
4 | 16 176 176 16
5 | 32 832 2112 832 32
6 | 64 3648 19328 19328 3648 64
7 | 128 15360 152448 309248 152448 15360 128
8 | 256 63232 1099008 3998464 3998464 1099008 63232 256
...
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T[n_, k_] := Sum[(-1)^(k-i)*Binomial[n+1, k-i]*(2*i)^(n), {i, 1, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten
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