A093563 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

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

\(P_{0}(x) = 1\)

\(P_{1}(x) = 7 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	1														8
2	14	10	1													25
3	28	44	14	1												87
4	56	160	100	19	1											336
5	112	536	560	195	25	1										1429
6	224	1720	2776	1535	345	32	1									6633
7	448	5384	12824	10451	3605	569	40	1								33322
8	896	16600	56680	65079	32081	7588	889	49	1							179863
9	1792	50696	243320	382075	257565	85197	14700	1330	59	1						1036735
10	3584	153880	1023976	2153695	1927465	853944	202797	26670	1920	70	1					6348002
11	7168	465224	4249784	11792451	13718485	7905073	2476320	442827	45870	2690	82	1				41105975
12	14336	1402840	17464360	63212039	94103361	69053996	27715633	6461763	901527	75460	3674	95	1			280409085
13	28672	4222856	71260280	333524555	627832205	577481333	290779060	85871500	15477033	1731587	119548	4909	109	1		2008333648
14	57344	12697240	289263976	1738883055	4100517785	4670201536	2903713813	1063622560	240641830	34524490	3166163	183365	6435	124	1	15057479717

Technical details

Node id: n-f258800410e48031 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-c9f0f8d386f8ac1c

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 Example

Triangle begins:
  0:    1;
  1:    7,     1;
  2:   14,    10,      1;
  3:   28,    44,     14,      1;
  4:   56,   160,    100,     19,      1;
  5:  112,   536,    560,    195,     25,     1;
  6:  224,  1720,   2776,   1535,    345,    32,     1;
  7:  448,  5384,  12824,  10451,   3605,   569,    40,    1;
  8:  896, 16600,  56680,  65079,  32081,  7588,   889,   49,  1;
  9: 1792, 50696, 243320, 382075, 257565, 85197, 14700, 1330, 59, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) + (n-1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] + (n - 1)*x*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] + (n - 1)*x*deriv(v[i - 1], x))/(n - 1)); 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]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from (6,1)-Pascal triangle.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.