A038305 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 + 390\,x + 144\,x^{2} + 234\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 9126\,x + 7956\,x^{2} + 1944\,x^{3} + 3042\,n x + 4212\,n x^{2} + 1296\,n x^{3} + 468\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 169 + 624\,x + 324\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	169	624	324															1117
1	2197	12168	12636	3456														30457
2	28561	210912	328536	179712	32400													780121
3	371293	3427320	7118280	5840640	2106000	279936												19143469
4	4826809	53466192	138806460	151856640	82134000	21835008	2286144											455211253
5	62748517	810903912	2526277572	3454738560	2491398000	993492864	208039104	17915904										10565514433
6	815730721	12047715264	43788811248	71858562048	64776348000	34441085952	10818033408	1863254016	136048896									240545589553
7	10604499373	176197835736	731898702288	1401241959936	1515766543200	1007401764096	421903302912	109000359936	15917720832	1007769600								5390940457909
8	137858491849	2545079849520	11893353912180	26023064970240	32841608436000	26192445866496	13711857344640	4723348930560	1034651854080	131010048000	7316407296							119241596110861
9	1792160394037	36394641848136	188972178826860	465162286343040	670907143764000	624253293151488	392159120056704	168859724267520	49318405044480	9367218432000	1046246243328	52242776064						2608284661147657
10	23298085122481	516142193482656	2947965989699016	8062812963279360	13082689303398000	13911930533090304	10196137121474304	5268423397146624	1923417796734720	487095358464000	81607206979584	8149873065984	367876214784					56510037698151817
11	302875106592253	7269002558214072	45291477478103064	136261539079421184	245663832474918000	293889532511532672	246163881932736768	148393925686296576	65011521529633536	20579778895104000	4597205993183232	688664274075648	62171080298496	2559896027136				1214177968496136637
12	3937376385699289	101766035814997008	686920741751229804	2254509101132241408	4471081751043507600	5943099435233216256	5600228313969761472	3858242067843710976	1972016153065550592	749103951781785600	209172872689837056	41778965960589312	5657568307163136	465901076938752	17631936921600			25897997867993149861
13	51185893014090757	1417455498851744040	10303811126268447060	36635772893398922880	79260085586680362000	115890438987047716992	121338280136011498560	94044650403690455040	54934735692540337920	24345878432908032000	8157742034903645184	2036724590578728960	367741939965603840	45425355001528320	3438227699712000	120367356051456		548833487165916877009
14	665416609183179841	19655382917410850688	153085193875988356320	586172366294382766080	1373841483502459608000	2191382846300538648576	2523836226829039170048	2173476364885290516480	1428303128006048785920	723420387720695808000	282801723876659699712	84727742968075124736	19122580878211399680	3149491280105963520	357575680770048000	25036410058702848	815300763254784	11564023763335681883233

Technical details

Node id: n-dc4135fd9023e99e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-53bee5cda939f818

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:           169,            624,             324;
  1:          2197,          12168,           12636,            3456;
  2:         28561,         210912,          328536,          179712,           32400;
  3:        371293,        3427320,         7118280,         5840640,         2106000,          279936;
  4:       4826809,       53466192,       138806460,       151856640,        82134000,        21835008,         2286144;
  5:      62748517,      810903912,      2526277572,      3454738560,      2491398000,       993492864,       208039104,        17915904;
  6:     815730721,    12047715264,     43788811248,     71858562048,     64776348000,     34441085952,     10818033408,      1863254016,      136048896;
  7:   10604499373,   176197835736,    731898702288,   1401241959936,   1515766543200,   1007401764096,    421903302912,    109000359936,    15917720832,    1007769600;
  8:  137858491849,  2545079849520,  11893353912180,  26023064970240,  32841608436000,  26192445866496,  13711857344640,   4723348930560,  1034651854080,  131010048000,    7316407296;
  9: 1792160394037, 36394641848136, 188972178826860, 465162286343040, 670907143764000, 624253293151488, 392159120056704, 168859724267520, 49318405044480, 9367218432000, 1046246243328, 52242776064;

%F Formula

The row polynomials satisfy (169+(234*n+390)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (2197+(3042*n+9126)*x+(468*n^2+4212*n+7956)*x^2+(216*n^2+1296*n+1944)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 624*x + 324*x^2.

%t Mathematica

P[0, x_] := 169 + 624*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2197 + (3042*n + 9126)*x + (468*n^2 + 4212*n + 7956)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*P[n - 1, x])/(169 + (234*n + 390)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 624*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (3042*n + 9126)*x + (468*n^2 + 4212*n + 7956)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*v[i - 1])/(169 + (234*n + 390)*x + (36*n^2 + 144*n + 144)*x^2)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.