A086645 → ⊙ C(n+1,k) (shift +2) → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + 3\,x + 3\,x^{2} + x^{3} + 3\,n + 9\,n x + 9\,n x^{2} + 3\,n x^{3} + 2\,n^{2} + 6\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n}(x) = \left(12 + 108\,x + 396\,x^{2} + 756\,x^{3} + 792\,x^{4} + 432\,x^{5} + 96\,x^{6} - 20\,n - 180\,n x - 660\,n x^{2} - 1260\,n x^{3} - 1320\,n x^{4} - 720\,n x^{5} - 160\,n x^{6} + 8\,n^{2} + 72\,n^{2} x + 264\,n^{2} x^{2} + 504\,n^{2} x^{3} + 528\,n^{2} x^{4} + 288\,n^{2} x^{5} + 64\,n^{2} x^{6}\right)\,P_{n-3}(x) + \left(-12 - 84\,x - 228\,x^{2} - 300\,x^{3} - 192\,x^{4} - 48\,x^{5} + 34\,n + 94\,n x + 70\,n x^{2} - 14\,n x^{3} - 32\,n x^{4} - 8\,n x^{5} - 36\,n^{2} - 36\,n^{2} x + 180\,n^{2} x^{2} + 396\,n^{2} x^{3} + 288\,n^{2} x^{4} + 72\,n^{2} x^{5}\right)\,P_{n-2}(x) + \left(-48\,x - 288\,x^{2} - 672\,x^{3} - 768\,x^{4} - 432\,x^{5} - 96\,x^{6}\right)\,P_{n-2}'(x) + \left(1 + 5\,x + 9\,x^{2} + 7\,x^{3} + 2\,x^{4} - 11\,n - 55\,n x - 99\,n x^{2} - 77\,n x^{3} - 22\,n x^{4} + 30\,n^{2} + 150\,n^{2} x + 270\,n^{2} x^{2} + 210\,n^{2} x^{3} + 60\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(48\,x + 192\,x^{2} + 288\,x^{3} + 192\,x^{4} + 48\,x^{5} - 54\,n x - 216\,n x^{2} - 324\,n x^{3} - 216\,n x^{4} - 54\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 14\,x + 14\,x^{2}\)
\(P_{2}(x) = 1 + 48\,x + 138\,x^{2} + 92\,x^{3}\)
\(P_{3}(x) = 1 + 116\,x + 762\,x^{2} + 1292\,x^{3} + 646\,x^{4}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 14 | 14 | 29 | |||||||
| 2 | 1 | 48 | 138 | 92 | 279 | ||||||
| 3 | 1 | 116 | 762 | 1292 | 646 | 2817 | |||||
| 4 | 1 | 230 | 3010 | 9760 | 11630 | 4652 | 29283 | ||||
| 5 | 1 | 402 | 9420 | 52160 | 111390 | 102372 | 34124 | 309869 | |||
| 6 | 1 | 644 | 24864 | 219800 | 748510 | 1186752 | 887348 | 253528 | 3321447 | ||
| 7 | 1 | 968 | 57708 | 774424 | 3941210 | 9589384 | 12070772 | 7606552 | 1901638 | 35942657 | |
| 8 | 1 | 1386 | 121212 | 2369136 | 17260362 | 60423804 | 113868804 | 118747008 | 64659798 | 14368844 | 391820355 |
Compute more
?
Technical details
- Node id
n-257453fbf28c3b7a· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-d122a18ed94aa5be
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1, 14, 14; 2: 1, 48, 138, 92; 3: 1, 116, 762, 1292, 646; 4: 1, 230, 3010, 9760, 11630, 4652; 5: 1, 402, 9420, 52160, 111390, 102372, 34124; 6: 1, 644, 24864, 219800, 748510, 1186752, 887348, 253528; 7: 1, 968, 57708, 774424, 3941210, 9589384, 12070772, 7606552, 1901638; 8: 1, 1386, 121212, 2369136, 17260362, 60423804, 113868804, 118747008, 64659798, 14368844; 9: 1, 1910, 235170, 6463920, 65276610, 315404964, 834345120, 1280286240, 1139132790, 546040820, 109208164; 10: 1, 2552, 427790, 16046580, 218802210, 1416460584, 5048017128, 10686883680, 13813789110, 10714557920, 4586896204, 833981128;
%F FormulaThe row polynomials satisfy (2*n^2+3*n+1+(6*n^2+9*n+3)*x+(6*n^2+9*n+3)*x^2+(2*n^2+3*n+1)*x^3)*P(n,x) = (8*n^2-20*n+12+(72*n^2-180*n+108)*x+(264*n^2-660*n+396)*x^2+(504*n^2-1260*n+756)*x^3+(528*n^2-1320*n+792)*x^4+(288*n^2-720*n+432)*x^5+(64*n^2-160*n+96)*x^6)*P(n-3,x) - (36*n^2-34*n+12+(36*n^2-94*n+84)*x-(180*n^2+70*n-228)*x^2-(396*n^2-14*n-300)*x^3-(288*n^2-32*n-192)*x^4-(72*n^2-8*n-48)*x^5)*P(n-2,x) - (48*x+288*x^2+672*x^3+768*x^4+432*x^5+96*x^6)*d/dx P(n-2,x) + (30*n^2-11*n+1+(150*n^2-55*n+5)*x+(270*n^2-99*n+9)*x^2+(210*n^2-77*n+7)*x^3+(60*n^2-22*n+2)*x^4)*P(n-1,x) - ((54*n-48)*x+(216*n-192)*x^2+(324*n-288)*x^3+(216*n-192)*x^4+(54*n-48)*x^5)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1 + 14*x + 14*x^2, P(2,x) = 1 + 48*x + 138*x^2 + 92*x^3, P(3,x) = 1 + 116*x + 762*x^2 + 1292*x^3 + 646*x^4.
%t MathematicaP[1, x_] := 1 + 14*x + 14*x^2;
P[2, x_] := 1 + 48*x + 138*x^2 + 92*x^3;
P[3, x_] := 1 + 116*x + 762*x^2 + 1292*x^3 + 646*x^4;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 20*n + 12 + (72*n^2 - 180*n + 108)*x + (264*n^2 - 660*n + 396)*x^2 + (504*n^2 - 1260*n + 756)*x^3 + (528*n^2 - 1320*n + 792)*x^4 + (288*n^2 - 720*n + 432)*x^5 + (64*n^2 - 160*n + 96)*x^6)*P[n - 3, x] - (36*n^2 - 34*n + 12 + (36*n^2 - 94*n + 84)*x - (180*n^2 + 70*n - 228)*x^2 - (396*n^2 - 14*n - 300)*x^3 - (288*n^2 - 32*n - 192)*x^4 - (72*n^2 - 8*n - 48)*x^5)*P[n - 2, x] - (48*x + 288*x^2 + 672*x^3 + 768*x^4 + 432*x^5 + 96*x^6)*D[P[n - 2, x], x] + (30*n^2 - 11*n + 1 + (150*n^2 - 55*n + 5)*x + (270*n^2 - 99*n + 9)*x^2 + (210*n^2 - 77*n + 7)*x^3 + (60*n^2 - 22*n + 2)*x^4)*P[n - 1, x] - ((54*n - 48)*x + (216*n - 192)*x^2 + (324*n - 288)*x^3 + (216*n - 192)*x^4 + (54*n - 48)*x^5)*D[P[n - 1, x], x])/(2*n^2 + 3*n + 1 + (6*n^2 + 9*n + 3)*x + (6*n^2 + 9*n + 3)*x^2 + (2*n^2 + 3*n + 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1 + 14*x + 14*x^2; v[2] = 1 + 48*x + 138*x^2 + 92*x^3; v[3] = 1 + 116*x + 762*x^2 + 1292*x^3 + 646*x^4; for(n = 4, N, my(i = n); v[i] = ((8*n^2 - 20*n + 12 + (72*n^2 - 180*n + 108)*x + (264*n^2 - 660*n + 396)*x^2 + (504*n^2 - 1260*n + 756)*x^3 + (528*n^2 - 1320*n + 792)*x^4 + (288*n^2 - 720*n + 432)*x^5 + (64*n^2 - 160*n + 96)*x^6)*v[i - 3] - (36*n^2 - 34*n + 12 + (36*n^2 - 94*n + 84)*x - (180*n^2 + 70*n - 228)*x^2 - (396*n^2 - 14*n - 300)*x^3 - (288*n^2 - 32*n - 192)*x^4 - (72*n^2 - 8*n - 48)*x^5)*v[i - 2] - (48*x + 288*x^2 + 672*x^3 + 768*x^4 + 432*x^5 + 96*x^6)*deriv(v[i - 2], x) + (30*n^2 - 11*n + 1 + (150*n^2 - 55*n + 5)*x + (270*n^2 - 99*n + 9)*x^2 + (210*n^2 - 77*n + 7)*x^3 + (60*n^2 - 22*n + 2)*x^4)*v[i - 1] - ((54*n - 48)*x + (216*n - 192)*x^2 + (324*n - 288)*x^3 + (216*n - 192)*x^4 + (54*n - 48)*x^5)*deriv(v[i - 1], x))/(2*n^2 + 3*n + 1 + (6*n^2 + 9*n + 3)*x + (6*n^2 + 9*n + 3)*x^2 + (2*n^2 + 3*n + 1)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [-1, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
- ↑ Pólya–Schur theorem on multiplier sequences (above)
-
Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
from
Triangle read by rows: T(n, k) = binomial(2n, 2k).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle read by rows: T(n, k) = binomial(2n, 2k).?
-
from
Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
- ↑ Pólya–Schur theorem on multiplier sequences (above)
- ↑ Pólya–Schur theorem on multiplier sequences (above)
-
from
Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
Related ?
Same fingerprint ?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023