Triangle of unsigned 2-Lah numbers.
A143497 (tail) ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A143497
\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(4 + 2\,x\right)\,P_{n-1}'(x) + \left(x\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 20 + 10\,x + x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 20 | 10 | 1 | 31 | |||||||
| 1 | 120 | 90 | 18 | 1 | 229 | ||||||
| 2 | 840 | 840 | 252 | 28 | 1 | 1961 | |||||
| 3 | 6720 | 8400 | 3360 | 560 | 40 | 1 | 19081 | ||||
| 4 | 60480 | 90720 | 45360 | 10080 | 1080 | 54 | 1 | 207775 | |||
| 5 | 604800 | 1058400 | 635040 | 176400 | 25200 | 1890 | 70 | 1 | 2501801 | ||
| 6 | 6652800 | 13305600 | 9313920 | 3104640 | 554400 | 55440 | 3080 | 88 | 1 | 32989969 | |
| 7 | 79833600 | 179625600 | 143700480 | 55883520 | 11975040 | 1496880 | 110880 | 4752 | 108 | 1 | 472630861 |
Compute more
?
Other recurrences
Image under 1+D · proved equal for all n (proof step)
\(P_{n}(x) = \left(5 + x + n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 20 + 10\,x + x^{2}\)
Technical details
- Node id
n-f3352d3240f9f08b· revision 2 · JSON- Definition source
oeis_catalog:A143497derived:one_plus_derivative(equal bye-b550228e4b1ac8f5)- Fingerprint
f6-a76d8ad579b60b88
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 2)
- 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 ?
Row sums ? A062147
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 20, 10, 1; 1: 120, 90, 18, 1; 2: 840, 840, 252, 28, 1; 3: 6720, 8400, 3360, 560, 40, 1; 4: 60480, 90720, 45360, 10080, 1080, 54, 1; 5: 604800, 1058400, 635040, 176400, 25200, 1890, 70, 1; 6: 6652800, 13305600, 9313920, 3104640, 554400, 55440, 3080, 88, 1; 7: 79833600, 179625600, 143700480, 55883520, 11975040, 1496880, 110880, 4752, 108, 1; 8: 1037836800, 2594592000, 2335132800, 1037836800, 259459200, 38918880, 3603600, 205920, 7020, 130, 1; 9: 14529715200, 39956716800, 39956716800, 19978358400, 5708102400, 998917920, 110990880, 7927920, 360360, 10010, 154, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (4+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 20 + 10*x + x^2.
%t MathematicaP[0, x_] := 20 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + (4 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 20 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + x)*v[i - 1] + (4 + 2*x)*deriv(v[i - 1], x) + 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 - 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 (−∞, 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)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
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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
Used in 6
- one_plus_derivative of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (derivative polynomial, shift 0)
- derivative of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (derivative polynomial, shift 0)
- shift_x_plus_one of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (affine, shift 0)
- reversal of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (reversal, shift 0)
- mobius_x_over_x_plus_one of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (mobius, shift 0)
- theta_plus_one of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (euler polynomial, shift 0)
Related ?
Same fingerprint ?
- A136124 → basis A008277 → D+1
- A143491 → basis A008277 → D
- A143497 → reverse
- A136124 → basis A008277 → D
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
- [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
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.