A088617 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-2 - 2\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(4 + 6\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	6	7	2													15
4	22	41	25	5												93
5	90	231	219	91	14											645
6	394	1289	1666	1064	336	42										4791
7	1806	7183	11780	10200	4920	1254	132									37275
8	8558	40081	79731	87345	56925	22077	4719	429								299865
9	41586	224143	524405	695695	572495	299299	97097	17875	1430							2474025
10	206098	1256465	3380740	5269880	5245240	3457454	1509508	420992	68068	4862						20819307
11	1037718	7059735	21477510	38480520	44968560	35818146	19695312	7383168	1806012	260338	16796					178003815
12	5293446	39753273	134930445	273241605	366832830	342842136	227631768	107379480	35271600	7684170	999362	58786				1541918901
13	27297738	224298231	840313551	1898128155	2879299500	3090172008	2406182856	1369724136	565779720	165392150	32481526	3848222	208012			13503125805
14	142078746	1267854873	5196728070	12954973680	21915622080	26568329436	23747740368	15846948480	7895121300	2900633670	763909212	136574736	14858000	742900		119352115551
15	745387038	7178461215	31954358280	87146576400	162678824880	219893080836	221958837672	169969607808	99230045772	43953544778	14541405736	3484498160	571735840	57500460	2674440	1063366539315

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(3 - n\right)\,P_{n-2}(x) + \left(-9 - 6\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 1\)

\(P_{2}(x) = 2 + x\)

Technical details

Node id: n-3ce0bead7a116750 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-907ff3ac724c18dd)
Fingerprint: f6-e4c3b23b17de5bef

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 Example

Triangle begins:
   1:      1;
   2:      2,       1;
   3:      6,       7,       2;
   4:     22,      41,      25,       5;
   5:     90,     231,     219,      91,      14;
   6:    394,    1289,    1666,    1064,     336,      42;
   7:   1806,    7183,   11780,   10200,    4920,    1254,     132;
   8:   8558,   40081,   79731,   87345,   56925,   22077,    4719,    429;
   9:  41586,  224143,  524405,  695695,  572495,  299299,   97097,  17875,  1430;
  10: 206098, 1256465, 3380740, 5269880, 5245240, 3457454, 1509508, 420992, 68068, 4862;

%F Formula

The row polynomials satisfy n*P(n,x) = (3*n-2+(2*n-2)*x)*P(n-1,x) + (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 2 + (2*n - 2)*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n - 2 + (2*n - 2)*x)*v[i - 1] + (4 + 6*x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - 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
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.?
  - ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A253284 → basis A007318

References

[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.