A038333 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 19\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 532\,x + 361\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	196	532	361															1089
1	2744	11172	15162	6859														35937
2	38416	208544	424536	384104	130321													1185921
3	537824	3649520	9905840	13443640	9122470	2476099												39135393
4	7529536	61311936	208022640	376421920	383143740	207992316	47045881											1291467969
5	105413504	1001428288	4077243744	9222337040	12516028840	10191623484	4610496338	893871739										42618442977
6	1475789056	16022852608	76108549888	206580349696	350448807520	380487276736	258187794928	100113634768	16983563041									1406408618241
7	20661046784	252359928576	1369953897984	4338187343616	8831309949504	11985349217184	10843887386976	6307158990384	2139928943166	322687697779								46411484401953
8	289254654976	3925598888960	23974193214720	86763746872320	206063898821760	335589778081152	379536058544160	294334086217920	149795026021620	45176277689060	6131066257801							1531578985264449
9	4049565169664	60454222889984	410225083896320	1670202127292160	4533405774078720	8613470970749568	11689710603160128	11331862319389920	7689478002443160	3478573382057620	944184203701354	116490258898219						50542106513726817
10	56693912375296	923300858683392	6891781409458176	31177106376120320	95201521255653120	206723303297989632	327311896888483584	380750573931501312	322958076102612720	194800109395226720	79311473110913736	19570363494900792	2213314919066161					1667889514952984961
11	793714773254144	14003396356698112	114027656047398912	567423336045389824	1925186318725429760	4702955150029264128	8510109319100573184	11549434075922206464	11755673970135103008	8863404977482815760	4811562702062099984	1780903078035972072	402823315270041302	42052983462257059				55040353993448503713
12	11112006825558016	211128129685602304	1862451715440848896	10110452169536036864	37733651847018423296	102419912156192863232	208497678317964043008	323384154125821780992	384018683024413364928	347445475117326377792	235766572401042899216	116352334431683508704	39476684896464047596	8242384758602383564	799006685782884121			1816331681783800622529
13	155568095557812224	3166921945284034560	30085758480198328320	176932912966880645120	720369717079442626560	2150818155280050127872	4864945827419161003520	8488834045802821751040	11520560490732400947840	12160591629106423222720	9902196040843801767072	6108497557663384206960	2763367942752483331720	865450399653250274220	167791404014405665410	15181127029874798299		59938945498865420543457
14	2177953337809371136	47292701049574916096	481372135683173253120	3048690192660097269760	13446901385482929029120	43798478798430111694848	108974786534189206478848	211277647362203563581440	322575693740507226539520	389138932131405543127040	369681985524835265970688	273660690583319612471808	154748604794139066576320	64620296507442687141760	18792637249613434525920	3400572454691954818976	288441413567621167681	1977985201462558877934081

Technical details

Node id: n-155538ecb7326369 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-b35a687a1de5c177

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:           196,            532,             361;
  1:          2744,          11172,           15162,             6859;
  2:         38416,         208544,          424536,           384104,           130321;
  3:        537824,        3649520,         9905840,         13443640,          9122470,          2476099;
  4:       7529536,       61311936,       208022640,        376421920,        383143740,        207992316,          47045881;
  5:     105413504,     1001428288,      4077243744,       9222337040,      12516028840,      10191623484,        4610496338,         893871739;
  6:    1475789056,    16022852608,     76108549888,     206580349696,     350448807520,     380487276736,      258187794928,      100113634768,      16983563041;
  7:   20661046784,   252359928576,   1369953897984,    4338187343616,    8831309949504,   11985349217184,    10843887386976,     6307158990384,    2139928943166,     322687697779;
  8:  289254654976,  3925598888960,  23974193214720,   86763746872320,  206063898821760,  335589778081152,   379536058544160,   294334086217920,  149795026021620,   45176277689060,   6131066257801;
  9: 4049565169664, 60454222889984, 410225083896320, 1670202127292160, 4533405774078720, 8613470970749568, 11689710603160128, 11331862319389920, 7689478002443160, 3478573382057620, 944184203701354, 116490258898219;

%F Formula

The row polynomials satisfy P(n,x) = (14+19*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 532*x + 361*x^2.

%t Mathematica

P[0, x_] := 196 + 532*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 19*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] = 196 + 532*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 19*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
    
    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
    
    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

Related ?

Same fingerprint ?

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.