A038334 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 150\,x + 144\,x^{2} + 90\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(500 + 5400\,x + 12240\,x^{2} + 7776\,x^{3} + 1800\,n x + 6480\,n x^{2} + 5184\,n x^{3} + 720\,n^{2} x^{2} + 864\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 400 + 3840\,x + 5184\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	400	3840	5184															9424
1	8000	115200	311040	221184														655424
2	160000	3072000	12441600	17694720	8294400													41662720
3	3200000	76800000	414720000	884736000	829440000	286654464												2495550464
4	64000000	1843200000	12441600000	35389440000	49766400000	34398535680	9364045824											143267221504
5	1280000000	43008000000	348364800000	1238630400000	2322432000000	2407897497600	1310966415360	293534171136										7966113284096
6	25600000000	983040000000	9289728000000	39636172800000	92897280000000	128421199872000	104877313228800	46965467381760	8916100448256									432011901730816
7	512000000000	22118400000000	238878720000000	1189085184000000	3344302080000000	5778953994240000	6292638793728000	4226892064358400	1604898080686080	264180754022400								22962460071034880
8	10240000000000	491520000000000	5971968000000000	33973862400000000	111476736000000000	231158159769600000	314631939686400000	281792804290560000	160489808068608000	52836150804480000	7671809096810496							1200504998116458496
9	204800000000000	10813440000000000	145981440000000000	934281216000000000	3503554560000000000	8475799191552000000	13843805346201600000	15498604235980800000	11769252591697920000	5811976588492800000	1687798001298309120	219122084616339456						61901193495839768576
10	4096000000000000	235929600000000000	3503554560000000000	24914165760000000000	105106636800000000000	290598829424640000000	553752213848064000000	743933003327078400000	706155155501875200000	464958127079424000000	202535760155797094400	52589300307921469440	6171938716693561344					3154458711081493725184
11	81920000000000000	5111808000000000000	82811289600000000000	647768309760000000000	3036413952000000000000	9444461956300800000000	20567939371499520000000	32237096810840064000000	36720068086097510400000	30222278260162560000000	17553099213502414848000	6836609040029791027200	1604704066340325949440	171791714339210133504				159130235798472196358144
12	1638400000000000000	110100480000000000000	1932263424000000000000	16488647884800000000000	85019590656000000000000	293827705307136000000000	719877878002483200000000	1289483872433602560000000	1713603177351217152000000	1692447582569103360000000	1228716944945169039360000	638083510402780495872000	224658569287645632921600	48101680014978837381120	4733037027712932249600			7957086198186629209784320
13	32768000000000000000	2359296000000000000000	44590694400000000000000	412216197120000000000000	2318716108800000000000000	8814831159214080000000000	23995929266749440000000000	48355645216260096000000000	73440136172195020800000000	84622379128455168000000000	73723016696710142361600000	47856263280208537190400000	22465856928764563292160000	7215252002246825607168000	1419911108313879674880000	129243464436747803295744		394816379487874500729503744
14	655360000000000000000	50331648000000000000000	1019215872000000000000000	10146860236800000000000000	61832429568000000000000000	256431451904409600000000000	767869736535982080000000000	1719311829911470080000000000	2937605446887800832000000000	3868451617300807680000000000	3931894223824540925952000000	3062800849933346380185600000	1797268554301165063372800000	769626880239661398097920000	227185777330220747980800000	41357908619759297054638080	3501690114583135795544064	19456355459588547220439302144

Technical details

Node id: n-1fffa59090d6ab2a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-dc876aec23f20348

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:             400,              3840,               5184;
  1:            8000,            115200,             311040,             221184;
  2:          160000,           3072000,           12441600,           17694720,             8294400;
  3:         3200000,          76800000,          414720000,          884736000,           829440000,           286654464;
  4:        64000000,        1843200000,        12441600000,        35389440000,         49766400000,         34398535680,           9364045824;
  5:      1280000000,       43008000000,       348364800000,      1238630400000,       2322432000000,       2407897497600,        1310966415360,         293534171136;
  6:     25600000000,      983040000000,      9289728000000,     39636172800000,      92897280000000,     128421199872000,      104877313228800,       46965467381760,        8916100448256;
  7:    512000000000,    22118400000000,    238878720000000,   1189085184000000,    3344302080000000,    5778953994240000,     6292638793728000,     4226892064358400,     1604898080686080,     264180754022400;
  8:  10240000000000,   491520000000000,   5971968000000000,  33973862400000000,  111476736000000000,  231158159769600000,   314631939686400000,   281792804290560000,   160489808068608000,   52836150804480000,    7671809096810496;
  9: 204800000000000, 10813440000000000, 145981440000000000, 934281216000000000, 3503554560000000000, 8475799191552000000, 13843805346201600000, 15498604235980800000, 11769252591697920000, 5811976588492800000, 1687798001298309120, 219122084616339456;

%F Formula

The row polynomials satisfy (25+(90*n+150)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (500+(1800*n+5400)*x+(720*n^2+6480*n+12240)*x^2+(864*n^2+5184*n+7776)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 3840*x + 5184*x^2.

%t Mathematica

P[0, x_] := 400 + 3840*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((500 + (1800*n + 5400)*x + (720*n^2 + 6480*n + 12240)*x^2 + (864*n^2 + 5184*n + 7776)*x^3)*P[n - 1, x])/(25 + (90*n + 150)*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] = 400 + 3840*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((500 + (1800*n + 5400)*x + (720*n^2 + 6480*n + 12240)*x^2 + (864*n^2 + 5184*n + 7776)*x^3)*v[i - 1])/(25 + (90*n + 150)*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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

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.