A048854 → x ↦ −2x (shift +2)
Proved for all n all roots real, in [0, ∞)alternating coefficient signs
Recurrence from A176231
\(P_{n}(x) = \left(-1 + x\right)\,P_{n-1}(x) + \left(2 - 4\,x\right)\,P_{n-1}'(x) + \left(4\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 3 - 6\,x + x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | x15 | x16 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 3 | -6 | 1 | -2 | ||||||||||||||
| 2 | -15 | 45 | -15 | 1 | 16 | |||||||||||||
| 3 | 105 | -420 | 210 | -28 | 1 | -132 | ||||||||||||
| 4 | -945 | 4725 | -3150 | 630 | -45 | 1 | 1216 | |||||||||||
| 5 | 10395 | -62370 | 51975 | -13860 | 1485 | -66 | 1 | -12440 | ||||||||||
| 6 | -135135 | 945945 | -945945 | 315315 | -45045 | 3003 | -91 | 1 | 138048 | |||||||||
| 7 | 2027025 | -16216200 | 18918900 | -7567560 | 1351350 | -120120 | 5460 | -120 | 1 | -1601264 | ||||||||
| 8 | -34459425 | 310134825 | -413513100 | 192972780 | -41351310 | 4594590 | -278460 | 9180 | -153 | 1 | 18108928 | |||||||
| 9 | 654729075 | -6547290750 | 9820936125 | -5237832600 | 1309458150 | -174594420 | 13226850 | -581400 | 14535 | -190 | 1 | -161934624 | ||||||
| 10 | -13749310575 | 151242416325 | -252070693875 | 151242416325 | -43212118950 | 6721885170 | -611080470 | 33575850 | -1119195 | 21945 | -231 | 1 | -404007680 | |||||
| 11 | 316234143225 | -3794809718700 | 6957151150950 | -4638100767300 | 1490818103775 | -265034329560 | 28109701620 | -1853386920 | 77224455 | -2018940 | 31878 | -276 | 1 | 92590134208 | ||||
| 12 | -7905853580625 | 102776096548125 | -205552193096250 | 150738274937250 | -53835098191875 | 10767019638375 | -1305093289500 | 100391791500 | -5019589575 | 164038875 | -3453450 | 44850 | -325 | 1 | -4221314202624 | |||
| 13 | 213458046676875 | -2988412653476250 | 6474894082531875 | -5179915266025500 | 2034966711652875 | -452214824811750 | 61665657928875 | -5421156741000 | 316234143225 | -12401338950 | 326351025 | -5651100 | 61425 | -378 | 1 | 159324751301248 | ||
| 14 | -6190283353629375 | 92854250304440625 | -216659917377028125 | 187771928393424375 | -80473683597181875 | 19671344879311125 | -2980506799895625 | 294775397791875 | -19651693186125 | 899097073875 | -28392539175 | 614557125 | -8906625 | 82215 | -435 | 1 | -5730872535686144 | |
| 15 | 191898783962510625 | -3070380543400170000 | 7675951358500425000 | -7164221267933730000 | 3326245588683517500 | -886998823648938000 | 147833137274823000 | -16245399700530000 | 1218404977539750 | -63707449806000 | 2347116571800 | -60964066800 | 1104421500 | -13592880 | 107880 | -496 | 1 | 205239818509082880 |
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Technical details
- Node id
n-aceeb2572b6504c7· revision 0 · JSON- Definition source
oeis_catalog:A176231- Fingerprint
f6-fa4a974805ea7cc4
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): no - 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: 3, -6, 1; 2: -15, 45, -15, 1; 3: 105, -420, 210, -28, 1; 4: -945, 4725, -3150, 630, -45, 1; 5: 10395, -62370, 51975, -13860, 1485, -66, 1; 6: -135135, 945945, -945945, 315315, -45045, 3003, -91, 1; 7: 2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1; 8: -34459425, 310134825, -413513100, 192972780, -41351310, 4594590, -278460, 9180, -153, 1; 9: 654729075, -6547290750, 9820936125, -5237832600, 1309458150, -174594420, 13226850, -581400, 14535, -190, 1; 10: -13749310575, 151242416325, -252070693875, 151242416325, -43212118950, 6721885170, -611080470, 33575850, -1119195, 21945, -231, 1;
%F FormulaThe row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) + (2-4*x)*d/dx P(n-1,x) + 4*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 3 - 6*x + x^2.
%t MathematicaP[1, x_] := 3 - 6*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x] + (2 - 4*x)*D[P[n - 1, x], x] + 4*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 3 - 6*x + x^2; for(n = 2, N, my(i = n); v[i] = -(1 - x)*v[i - 1] + (2 - 4*x)*deriv(v[i - 1], x) + 4*x*deriv(deriv(v[i - 1], x), x)); 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 [0, +∞)
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n(-x) have one sign
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].?
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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
Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].?
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.