A272866 → Veronese 2 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + 2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(12 + 96\,x + 306\,x^{2} + 502\,x^{3} + 450\,x^{4} + 210\,x^{5} + 40\,x^{6} - 21\,n - 152\,n x - 460\,n x^{2} - 738\,n x^{3} - 659\,n x^{4} - 310\,n x^{5} - 60\,n x^{6} + 9\,n^{2} + 56\,n^{2} x + 154\,n^{2} x^{2} + 236\,n^{2} x^{3} + 209\,n^{2} x^{4} + 100\,n^{2} x^{5} + 20\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(-2 + 42\,x + 318\,x^{2} + 892\,x^{3} + 1298\,x^{4} + 1050\,x^{5} + 450\,x^{6} + 80\,x^{7} - 20\,n x - 140\,n x^{2} - 400\,n x^{3} - 600\,n x^{4} - 500\,n x^{5} - 220\,n x^{6} - 40\,n x^{7}\right)\,P_{n-1}'(x) + \left(-4\,x + 80\,x^{3} + 260\,x^{4} + 380\,x^{5} + 296\,x^{6} + 120\,x^{7} + 20\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 1 | 13 | 13 | 27 | |||||
| 3 | 1 | 33 | 93 | 62 | 189 | ||||
| 4 | 1 | 62 | 375 | 626 | 313 | 1377 | |||
| 5 | 1 | 100 | 1075 | 3320 | 3905 | 1562 | 9963 | ||
| 6 | 1 | 147 | 2490 | 12499 | 25782 | 23439 | 7813 | 72171 | |
| 7 | 1 | 203 | 4998 | 37513 | 119630 | 184569 | 136717 | 39062 | 522693 |
Compute more
?
Technical details
- Node id
n-efb049ae8c6d52aa· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-bd43e5bf90f88fc7
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 2; 2: 1, 13, 13; 3: 1, 33, 93, 62; 4: 1, 62, 375, 626, 313; 5: 1, 100, 1075, 3320, 3905, 1562; 6: 1, 147, 2490, 12499, 25782, 23439, 7813; 7: 1, 203, 4998, 37513, 119630, 184569, 136717, 39062; 8: 1, 268, 9058, 95704, 438475, 1015624, 1250002, 781252, 195313; 9: 1, 342, 15210, 216132, 1352691, 4386294, 7946778, 8129880, 4394529, 976562;
%F FormulaThe row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (9*n^2-21*n+12+(56*n^2-152*n+96)*x+(154*n^2-460*n+306)*x^2+(236*n^2-738*n+502)*x^3+(209*n^2-659*n+450)*x^4+(100*n^2-310*n+210)*x^5+(20*n^2-60*n+40)*x^6)*P(n-1,x) - (2+(20*n-42)*x+(140*n-318)*x^2+(400*n-892)*x^3+(600*n-1298)*x^4+(500*n-1050)*x^5+(220*n-450)*x^6+(40*n-80)*x^7)*d/dx P(n-1,x) - (4*x-80*x^3-260*x^4-380*x^5-296*x^6-120*x^7-20*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 21*n + 12 + (56*n^2 - 152*n + 96)*x + (154*n^2 - 460*n + 306)*x^2 + (236*n^2 - 738*n + 502)*x^3 + (209*n^2 - 659*n + 450)*x^4 + (100*n^2 - 310*n + 210)*x^5 + (20*n^2 - 60*n + 40)*x^6)*P[n - 1, x] - (2 + (20*n - 42)*x + (140*n - 318)*x^2 + (400*n - 892)*x^3 + (600*n - 1298)*x^4 + (500*n - 1050)*x^5 + (220*n - 450)*x^6 + (40*n - 80)*x^7)*D[P[n - 1, x], x] - (4*x - 80*x^3 - 260*x^4 - 380*x^5 - 296*x^6 - 120*x^7 - 20*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^2 - 21*n + 12 + (56*n^2 - 152*n + 96)*x + (154*n^2 - 460*n + 306)*x^2 + (236*n^2 - 738*n + 502)*x^3 + (209*n^2 - 659*n + 450)*x^4 + (100*n^2 - 310*n + 210)*x^5 + (20*n^2 - 60*n + 40)*x^6)*v[i - 1] - (2 + (20*n - 42)*x + (140*n - 318)*x^2 + (400*n - 892)*x^3 + (600*n - 1298)*x^4 + (500*n - 1050)*x^5 + (220*n - 450)*x^6 + (40*n - 80)*x^7)*deriv(v[i - 1], x) - (4*x - 80*x^3 - 260*x^4 - 380*x^5 - 296*x^6 - 120*x^7 - 20*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
-
Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
-
-
from
veronese_section_2_0 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 …?
Related ?
Same fingerprint ?
References
- [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.