A038271 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 5\,n x\right)\,P_{n}(x) = \left(49 + 70\,x + 25\,x^{2} + 35\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	10														17
2	49	140	75													264
3	343	1470	1575	500												3888
4	2401	13720	22050	14000	3125											55296
5	16807	120050	257250	245000	109375	18750										767232
6	117649	1008420	2701125	3430000	2296875	787500	109375									10450944
7	823543	8235430	26471025	42017500	37515625	19293750	5359375	625000								140341248
8	5764801	65883440	247062900	470596000	525218750	360150000	150062500	35000000	3515625							1863254016
9	40353607	518832090	2223566100	4941258000	6617756250	5672362500	3151312500	1102500000	221484375	19531250						24508956672
10	282475249	4035360700	19456203375	49412580000	77207156250	79413075000	55147968750	25725000000	7751953125	1367187500	107421875					319906381824
11	1977326743	31072277390	166458628875	475596082500	849278718750	1019134462500	849278718750	495206250000	198966796875	52636718750	8271484375	585937500				4148463403008
12	13841287201	237279209160	1398252482550	4438896770000	8917426546875	12229613550000	11889902062500	8319465000000	4178302734375	1473828125000	347402343750	49218750000	3173828125			53496602689536
13	96889010407	1799367336130	11567361446550	40393960607000	90165090640625	139111854131250	154568726812500	126178552500000	76045109765625	33529589843750	10537871093750	2239453125000	288818359375	17089843750		686539734515712
14	678223072849	13564461456980	94466785146825	359873467226000	883617888278125	1514773522762500	1893466903453125	1766499735000000	1242070126171875	657179960937500	258177841796875	73155468750000	14152099609375	1674804687500	91552734375	8773442841083904

Technical details

Node id: n-6bc4c10117d498cf · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9d686d1207b6ad5d

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:        7,        10;
  2:       49,       140,         75;
  3:      343,      1470,       1575,        500;
  4:     2401,     13720,      22050,      14000,       3125;
  5:    16807,    120050,     257250,     245000,     109375,      18750;
  6:   117649,   1008420,    2701125,    3430000,    2296875,     787500,     109375;
  7:   823543,   8235430,   26471025,   42017500,   37515625,   19293750,    5359375,     625000;
  8:  5764801,  65883440,  247062900,  470596000,  525218750,  360150000,  150062500,   35000000,   3515625;
  9: 40353607, 518832090, 2223566100, 4941258000, 6617756250, 5672362500, 3151312500, 1102500000, 221484375, 19531250;

%F Formula

The row polynomials satisfy (7+5*n*x)*P(n,x) = (49+(35*n+70)*x+(25*n+25)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((49 + (35*n + 70)*x + (25*n + 25)*x^2)*P[n - 1, x])/(7 + 5*n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((49 + (35*n + 70)*x + (25*n + 25)*x^2)*v[i - 1])/(7 + 5*n*x)); 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 Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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.