A038331 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(37 + 40\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1369 + 2960\,x + 1600\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1369	2960	1600								5929
1	50653	164280	177600	64000							456533
2	1874161	8104480	13142400	9472000	2560000						35153041
3	69343957	374832200	810448000	876160000	473600000	102400000					2706784157
4	2565726409	16642549680	44979864000	64835840000	52569600000	22732800000	4096000000				208422380089
5	94931877133	718403394520	2329956955200	4198120640000	4538508800000	2943897600000	1060864000000	163840000000			16048523266853
6	3512479453921	30378200682560	114944543123200	248528741888000	335849651200000	290464563200000	157007872000000	48496640000000	6553600000000		1235736291547681
7	129961739795077	1264492603411560	5468076122860800	13793345174784000	22367586769920000	24181174886400000	17427873792000000	8074690560000000	2182348800000000	262144000000000	95151694449171437

All 15 rows

Technical details

Node id: n-3de3dd57fdba145b · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-4f2c8cb4b6021dc1

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:               1369,                2960,                 1600;
  1:              50653,              164280,               177600,                64000;
  2:            1874161,             8104480,             13142400,              9472000,              2560000;
  3:           69343957,           374832200,            810448000,            876160000,            473600000,             102400000;
  4:         2565726409,         16642549680,          44979864000,          64835840000,          52569600000,           22732800000,            4096000000;
  5:        94931877133,        718403394520,        2329956955200,        4198120640000,        4538508800000,         2943897600000,         1060864000000,          163840000000;
  6:      3512479453921,      30378200682560,      114944543123200,      248528741888000,      335849651200000,       290464563200000,       157007872000000,        48496640000000,        6553600000000;
  7:    129961739795077,    1264492603411560,     5468076122860800,    13793345174784000,    22367586769920000,     24181174886400000,     17427873792000000,      8074690560000000,     2182348800000000,      262144000000000;
  8:   4808584372417849,   51984695918030800,   252898520682312000,   729076816381440000,  1379334517478400000,   1789406941593600000,   1612078325760000000,    995878502400000000,   403734528000000000,    96993280000000000,   10485760000000000;
  9: 177917621779460413, 2115777123863853560, 11436633101966776000, 37091783033405760000, 80198449801958400000, 121381437538099200000, 131223175716864000000, 101330637619200000000, 54773317632000000000, 19738132480000000000, 4267704320000000000, 419430400000000000;

%F Formula

The row polynomials satisfy P(n,x) = (37+40*x)*P(n-1,x) for n >= 1, with P(0,x) = 1369 + 2960*x + 1600*x^2.

%t Mathematica

P[0, x_] := 1369 + 2960*x + 1600*x^2;
P[n_, x_] := P[n, x] = Expand[(37 + 40*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1369 + 2960*x + 1600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (37 + 40*x)*v[i - 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    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
    
    Parent-row provers of polytool polytool ? 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.