A122075 Coefficients of a generalized Pell-Lucas polynomial read by rows.
1, 2, 1, 3, 3, 1, 5, 7, 4, 1, 8, 15, 12, 5, 1, 13, 30, 31, 18, 6, 1, 21, 58, 73, 54, 25, 7, 1, 34, 109, 162, 145, 85, 33, 8, 1, 55, 201, 344, 361, 255, 125, 42, 9, 1, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 144, 655, 1416, 1918, 1806, 1239, 630, 236, 63, 11, 1
Offset: 0
Examples
Triangle begins: 1 2 1 3 3 1 5 7 4 1 8 15 12 5 1 13 30 31 18 6 1 A055830 = (1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, ...) begins: 1 1, 0 2, 1, 0 3, 3, 1, 0 5, 7, 4, 1, 0 8, 15, 12, 5, 1, 0 13, 30, 31, 18, 6, 1, 0
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
- Tian-Xiao He and Peter J.-S. Shiue, Identities for linear recursive sequences of order 2, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.
- Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
- Y. Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370.
Crossrefs
See A055830 for another version.
Programs
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]; v[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%] (* A122075 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A037027 *) (* Clark Kimberling, Mar 05 2012 *) CoefficientList[CoefficientList[Series[-(1 + x)/(-1 + x*y + x + x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Dec 24 2017 *) -
PARI
T(n,k)={ sum(j=0,n-k+1, binomial(n-k-j+1,j)*binomial(n-j,k)) ; } { nmax=10 ; for(n=0,nmax, for(k=0,n, print1(T(n,k),",") ; ); ); }
Formula
T(n,k) = Sum_{j=0..n-k+1} binomial(n-k-j+1,j)*binomial(n-j,k).
Sum_{k>=0} T(n-k,k) = 2^n.
Sum_{k>=0} (-1)^k*T(n-k,k) = 2-delta(0,n).
G.f.: -(1+x)/(-1+x*y+x+x^2). - R. J. Mathar, Aug 11 2015
Comments