A193727 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 6\,x + 7\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	3														6
2	21	39	18													78
3	147	399	360	108												1014
4	1029	3675	4914	2916	648											13182
5	7203	31899	56448	49896	22032	3888										171366
6	50421	266511	586530	687960	453600	159408	23328									2227758
7	352947	2168103	5704776	8334900	7302960	3837456	1119744	139968								28960854
8	2470629	17294403	52942050	92572956	101130120	70679952	30862944	7698240	839808							376491102
9	17294403	135884595	474360768	965662992	1263348576	1101540384	640120320	239065344	52068096	5038848						4894384326
10	121060821	1054958583	4135832946	9605805552	14637417984	15290874144	11090084544	5514179328	1798868736	347680512	30233088					63626996238
11	847425747	8111075007	35280582120	92055636540	160096759200	194860626912	169375836672	105139762560	45677157120	13226976000	2297714688	181398528				827150951094
12	5931980229	61862079531	295630524882	856072948500	1673011133640	2324604943584	2354794618176	1752233357952	950578675200	366651774720	95445858816	15056077824	1088391168			10752962364222
13	41523861603	468626438091	2440586151360	7766293788792	16847515626480	26310301406928	30431191988736	26394401214720	17167450874112	8270034474240	2868031660032	678067697664	97955205120	6530347008		139788510734886
14	290667031221	3529528236255	19895861688066	69007573429704	164530372118112	285257203607376	370880152362720	367347960435456	278538563407104	160894946564352	69696428465664	21954663843840	4754092621824	633443659776	39182082048	1817250639553518

Technical details

Node id: n-4a277cbc3c10d12c · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-37fc5a8207304837

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        3,         3;
  2:       21,        39,        18;
  3:      147,       399,       360,       108;
  4:     1029,      3675,      4914,      2916,        648;
  5:     7203,     31899,     56448,     49896,      22032,       3888;
  6:    50421,    266511,    586530,    687960,     453600,     159408,     23328;
  7:   352947,   2168103,   5704776,   8334900,    7302960,    3837456,   1119744,    139968;
  8:  2470629,  17294403,  52942050,  92572956,  101130120,   70679952,  30862944,   7698240,   839808;
  9: 17294403, 135884595, 474360768, 965662992, 1263348576, 1101540384, 640120320, 239065344, 52068096, 5038848;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(6*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (6*n - 6)*x)*P[n - 1, 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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (6*n - 6)*x)*v[i - 1])/(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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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 Mirror of the triangle A193726.?
  - 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
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.