A157277
Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 2, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 10, 1, 1, 39, 39, 1, 1, 128, 470, 128, 1, 1, 397, 3558, 3558, 397, 1, 1, 1206, 22387, 55452, 22387, 1206, 1, 1, 3635, 128377, 632343, 632343, 128377, 3635, 1, 1, 10924, 698788, 6107192, 12269406, 6107192, 698788, 10924, 1, 1, 32793, 3686880, 53375112, 187721254, 187721254, 53375112, 3686880, 32793, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 10, 1;
1, 39, 39, 1;
1, 128, 470, 128, 1;
1, 397, 3558, 3558, 397, 1;
1, 1206, 22387, 55452, 22387, 1206, 1;
1, 3635, 128377, 632343, 632343, 128377, 3635, 1;
1, 10924, 698788, 6107192, 12269406, 6107192, 698788, 10924, 1;
Cf.
A157147,
A157148,
A157149,
A157150,
A157151,
A157152,
A157153,
A157154,
A157155,
A157156,
A157207,
A157208,
A157209,
A157210,
A157211,
A157212,
A157268,
A157272,
A157273,
A157274.
-
f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 05 2022 *)
-
def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n,k,m): # A157277
if (k==0 or k==n): return 1
else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022
A157278
Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 3, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 14, 1, 1, 69, 69, 1, 1, 292, 1134, 292, 1, 1, 1187, 11686, 11686, 1187, 1, 1, 4770, 100737, 254132, 100737, 4770, 1, 1, 19105, 795723, 4061249, 4061249, 795723, 19105, 1, 1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 14, 1;
1, 69, 69, 1;
1, 292, 1134, 292, 1;
1, 1187, 11686, 11686, 1187, 1;
1, 4770, 100737, 254132, 100737, 4770, 1;
1, 19105, 795723, 4061249, 4061249, 795723, 19105, 1;
1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 1;
Cf.
A157147,
A157148,
A157149,
A157150,
A157151,
A157152,
A157153,
A157154,
A157155,
A157156,
A157207,
A157208,
A157209,
A157210,
A157211,
A157212,
A157268,
A157272,
A157273,
A157274.
-
f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] + m*f[n,k]*T[n-2,k-1,m]];
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 06 2022 *)
-
def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n,k,m): # A157278
if (k==0 or k==n): return 1
else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) + m*f(n,k)*T(n-2,k-1,m)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 06 2022
A168518
Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 12, 1, 1, 51, 51, 1, 1, 170, 514, 170, 1, 1, 521, 3646, 3646, 521, 1, 1, 1552, 22247, 49472, 22247, 1552, 1, 1, 4591, 125565, 534995, 534995, 125565, 4591, 1, 1, 13590, 677776, 5058698, 9506078, 5058698, 677776, 13590, 1, 1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 12, 1;
1, 51, 51, 1;
1, 170, 514, 170, 1;
1, 521, 3646, 3646, 521, 1;
1, 1552, 22247, 49472, 22247, 1552, 1;
1, 4591, 125565, 534995, 534995, 125565, 4591, 1;
1, 13590, 677776, 5058698, 9506078, 5058698, 677776, 13590, 1;
1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1;
-
p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
Table[CoefficientList[p[x,n,-4,2,2], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)
-
def A168518(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168518(n,k,-4,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022