A290596 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(12 + 6\,x\right)\,P_{n-1}'(x) + \left(9 + 9\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 3 | 1 | 4 | ||||||
| 3 | 21 | 12 | 1 | 34 | |||||
| 4 | 225 | 171 | 27 | 1 | 424 | ||||
| 5 | 3213 | 2952 | 666 | 48 | 1 | 6880 | |||
| 6 | 57051 | 60345 | 17370 | 1830 | 75 | 1 | 136672 | ||
| 7 | 1207953 | 1428516 | 493695 | 67680 | 4095 | 108 | 1 | 3202048 | |
| 8 | 29652561 | 38454507 | 15367401 | 2573235 | 204435 | 8001 | 147 | 1 | 86260288 |
Compute more
?
Technical details
- Node id
n-42681aa94fdb77b6· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-3847c4c478bf2769
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 -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; 2: 3, 1; 3: 21, 12, 1; 4: 225, 171, 27, 1; 5: 3213, 2952, 666, 48, 1; 6: 57051, 60345, 17370, 1830, 75, 1; 7: 1207953, 1428516, 493695, 67680, 4095, 108, 1; 8: 29652561, 38454507, 15367401, 2573235, 204435, 8001, 147, 1; 9: 827024985, 1160128656, 522635652, 102737376, 10052910, 519120, 14196, 192, 1; 10: 25810060563, 38766698577, 19331741556, 4341816108, 502587666, 32111478, 1161972, 23436, 243, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (12+6*x)*d/dx P(n-1,x) + (9+9*x)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + (12 + 6*x)*D[P[n - 1, x], x] + (9 + 9*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] = 1; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + (12 + 6*x)*deriv(v[i - 1], x) + (9 + 9*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 (−∞, -1]
- 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
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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[3,1].?
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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[3,1].?
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.