A159854 → 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

\(\left(1 + 15\,x + 48\,x^{2} + 48\,x^{3} + 16\,x^{4} + 3\,n x + 20\,n x^{2} + 36\,n x^{3} + 16\,n x^{4} + n^{2} x^{2} + 4\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(2 + 38\,x + 174\,x^{2} + 314\,x^{3} + 248\,x^{4} + 72\,x^{5} + 6\,n x + 50\,n x^{2} + 132\,n x^{3} + 136\,n x^{4} + 48\,n x^{5} + 2\,n^{2} x^{2} + 10\,n^{2} x^{3} + 16\,n^{2} x^{4} + 8\,n^{2} x^{5}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 16\,x + 36\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	16	36															53
1	2	40	144	128														314
2	4	96	468	768	400													1736
3	8	224	1368	3200	3200	1152												9152
4	16	512	3744	11264	16400	11520	3136											46592
5	32	1152	9792	35840	68000	70272	37632	8192										230912
6	64	2560	24768	106496	248000	336384	266560	114688	20736									1120256
7	128	5632	61056	301056	828800	1387008	1448832	925696	331776	51200								5341184
8	256	12288	147456	819200	2598400	5160960	6673408	5636096	3006720	921600	123904							25100288
9	512	26624	350208	2162688	7756800	17805312	27396096	28704768	20279808	9267200	2478080	294912						116523008
10	1024	57344	820224	5570560	22272000	57950208	103262208	128974848	113218560	68608000	27382784	6488064	692224					535298048
11	2048	122880	1898496	14057472	61952000	180043776	364277760	527695872	552904704	416768000	220796928	78151680	16613376	1605632				2436890624
12	4096	262144	4349952	34865152	167833600	538509312	1218674688	2006974464	2441539584	2198732800	1450172416	681836544	216666112	41746432	3686400			11005853696
13	8192	557056	9879552	85196800	444620800	1560379392	3903291392	7197425664	9963233280	10425958400	8221278208	4815323136	2033754112	586055680	103219200	8388608		49358569472
14	16384	1179648	22265856	205520896	1155481600	4401266688	12054282240	24591204352	38144950272	45452492800	41673375744	29198647296	15370141696	5889458176	1551974400	251658240	18939904	219982856192

Technical details

Node id: n-9305fa2263e02428 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-8d0a5745cf4269c7

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:   1,    16,     36;
  1:   2,    40,    144,     128;
  2:   4,    96,    468,     768,     400;
  3:   8,   224,   1368,    3200,    3200,     1152;
  4:  16,   512,   3744,   11264,   16400,    11520,     3136;
  5:  32,  1152,   9792,   35840,   68000,    70272,    37632,     8192;
  6:  64,  2560,  24768,  106496,  248000,   336384,   266560,   114688,    20736;
  7: 128,  5632,  61056,  301056,  828800,  1387008,  1448832,   925696,   331776,   51200;
  8: 256, 12288, 147456,  819200, 2598400,  5160960,  6673408,  5636096,  3006720,  921600,  123904;
  9: 512, 26624, 350208, 2162688, 7756800, 17805312, 27396096, 28704768, 20279808, 9267200, 2478080, 294912;

%F Formula

The row polynomials satisfy (1+(3*n+15)*x+(n^2+20*n+48)*x^2+(4*n^2+36*n+48)*x^3+(4*n^2+16*n+16)*x^4)*P(n,x) = (2+(6*n+38)*x+(2*n^2+50*n+174)*x^2+(10*n^2+132*n+314)*x^3+(16*n^2+136*n+248)*x^4+(8*n^2+48*n+72)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 16*x + 36*x^2.

%t Mathematica

P[0, x_] := 1 + 16*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n + 38)*x + (2*n^2 + 50*n + 174)*x^2 + (10*n^2 + 132*n + 314)*x^3 + (16*n^2 + 136*n + 248)*x^4 + (8*n^2 + 48*n + 72)*x^5)*P[n - 1, x])/(1 + (3*n + 15)*x + (n^2 + 20*n + 48)*x^2 + (4*n^2 + 36*n + 48)*x^3 + (4*n^2 + 16*n + 16)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 16*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (6*n + 38)*x + (2*n^2 + 50*n + 174)*x^2 + (10*n^2 + 132*n + 314)*x^3 + (16*n^2 + 136*n + 248)*x^4 + (8*n^2 + 48*n + 72)*x^5)*v[i - 1])/(1 + (3*n + 15)*x + (n^2 + 20*n + 48)*x^2 + (4*n^2 + 36*n + 48)*x^3 + (4*n^2 + 16*n + 16)*x^4)); 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 Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    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

Related ?

Same fingerprint ?

A391533 → reverse → (θ+1)²

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.