A176230 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 21\,x + 40\,x^{2} + 30\,x^{3} + 8\,x^{4} - 2\,n - 22\,n x - 50\,n x^{2} - 42\,n x^{3} - 12\,n x^{4} + 4\,n^{2} x + 12\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(2 + 28\,x + 88\,x^{2} + 116\,x^{3} + 70\,x^{4} + 16\,x^{5} - 8\,n x - 32\,n x^{2} - 48\,n x^{3} - 32\,n x^{4} - 8\,n x^{5}\right)\,P_{n-1}'(x) + \left(4\,x + 20\,x^{2} + 40\,x^{3} + 40\,x^{4} + 20\,x^{5} + 4\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 3 | 12 | 10 | 25 | |||||
| 3 | 15 | 90 | 150 | 76 | 331 | ||||
| 4 | 105 | 840 | 2100 | 2128 | 764 | 5937 | |||
| 5 | 945 | 9450 | 31500 | 47880 | 34380 | 9496 | 133651 | ||
| 6 | 10395 | 124740 | 519750 | 1053360 | 1134540 | 626736 | 140152 | 3609673 | |
| 7 | 135135 | 1891890 | 9459450 | 23963940 | 34414380 | 28516488 | 12753832 | 2390480 | 113525595 |
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Technical details
- Node id
n-e2163acca2adac55· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-9d419c98ea340a64
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: 3, 12, 10; 3: 15, 90, 150, 76; 4: 105, 840, 2100, 2128, 764; 5: 945, 9450, 31500, 47880, 34380, 9496; 6: 10395, 124740, 519750, 1053360, 1134540, 626736, 140152; 7: 135135, 1891890, 9459450, 23963940, 34414380, 28516488, 12753832, 2390480; 8: 2027025, 32432400, 189189000, 575134560, 1032431400, 1140659520, 765229920, 286857600, 46206736; 9: 34459425, 620269650, 4135131000, 14665931280, 31592400840, 43630226640, 39026725920, 21944606400, 7069630608, 997313824;
%F FormulaThe row polynomials satisfy (1+x)*P(n,x) = -(2*n-3-(4*n^2-22*n+21)*x-(12*n^2-50*n+40)*x^2-(12*n^2-42*n+30)*x^3-(4*n^2-12*n+8)*x^4)*P(n-1,x) + (2-(8*n-28)*x-(32*n-88)*x^2-(48*n-116)*x^3-(32*n-70)*x^4-(8*n-16)*x^5)*d/dx P(n-1,x) + (4*x+20*x^2+40*x^3+40*x^4+20*x^5+4*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 3 - (4*n^2 - 22*n + 21)*x - (12*n^2 - 50*n + 40)*x^2 - (12*n^2 - 42*n + 30)*x^3 - (4*n^2 - 12*n + 8)*x^4)*P[n - 1, x] + (2 - (8*n - 28)*x - (32*n - 88)*x^2 - (48*n - 116)*x^3 - (32*n - 70)*x^4 - (8*n - 16)*x^5)*D[P[n - 1, x], x] + (4*x + 20*x^2 + 40*x^3 + 40*x^4 + 20*x^5 + 4*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 3 - (4*n^2 - 22*n + 21)*x - (12*n^2 - 50*n + 40)*x^2 - (12*n^2 - 42*n + 30)*x^3 - (4*n^2 - 12*n + 8)*x^4)*v[i - 1] + (2 - (8*n - 28)*x - (32*n - 88)*x^2 - (48*n - 116)*x^3 - (32*n - 70)*x^4 - (8*n - 16)*x^5)*deriv(v[i - 1], x) + (4*x + 20*x^2 + 40*x^3 + 40*x^4 + 20*x^5 + 4*x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); 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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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from
Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].?
References
- [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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.