A193726 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 20\,x + 7\,n + 20\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 8\,x\)

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	8														11
2	21	116	160													297
3	147	1232	3440	3200												8019
4	1029	11564	48720	91200	64000											216513
5	7203	101528	572320	1612800	2272000	1280000										5845851
6	50421	854756	6036800	22736000	48160000	54400000	25600000									157837977
7	352947	6991712	59352720	279888000	791840000	1344000000	1267200000	512000000								4261625379
8	2470629	56000924	555303280	3146270400	11140640000	25244800000	35750400000	28928000000	10240000000							115063885233
9	17294403	441419048	5007141440	33129958400	140909888000	399526400000	755148800000	917504000000	650240000000	204800000000						3106724901291
10	121060821	3435821396	43878371040	332052537600	1648968384000	5614882560000	13276569600000	21525504000000	22901760000000	14438400000000	4096000000000					83881572334857
11	847425747	26471966192	375865025200	3201935184000	18183829440000	72283545600000	205233638400000	416209920000000	590822400000000	559104000000000	317440000000000	81920000000000				2264802453041139
12	5931980229	202252278284	3160494500240	29930846792000	191325509760000	869661408000000	2882306380800000	7018142208000000	12459955200000000	15730176000000000	13404160000000000	6922240000000000	1638400000000000			61149666232110753
13	41523861603	1534405552568	26168507067360	272725817548800	1937895504160000	9914140051200000	37569372825600000	106773123072000000	227582530560000000	359310336000000000	408432640000000000	316538880000000000	149913600000000000	32768000000000000		1651040988266990331
14	290667031221	11571316100036	213867660522880	2432450864188800	19019784880096000	108156890441600000	461268410803200000	1498799318016000000	3728540175360000000	7066822963200000000	10045235200000000000	10384424960000000000	7380172800000000000	3227648000000000000	655360000000000000	44578106683208738937

Technical details

Node id: n-7ffe75f198ff0f55 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-918ab47be015a3a4

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,         8;
  2:       21,       116,        160;
  3:      147,      1232,       3440,        3200;
  4:     1029,     11564,      48720,       91200,        64000;
  5:     7203,    101528,     572320,     1612800,      2272000,      1280000;
  6:    50421,    854756,    6036800,    22736000,     48160000,     54400000,     25600000;
  7:   352947,   6991712,   59352720,   279888000,    791840000,   1344000000,   1267200000,    512000000;
  8:  2470629,  56000924,  555303280,  3146270400,  11140640000,  25244800000,  35750400000,  28928000000,  10240000000;
  9: 17294403, 441419048, 5007141440, 33129958400, 140909888000, 399526400000, 755148800000, 917504000000, 650240000000, 204800000000;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(20*n-20)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (20*n - 20)*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (20*n - 20)*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 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.?
  - 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.