A193729 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 16\,x + 7\,n + 16\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	8														11
2	21	104	128													253
3	147	1064	2560	2048												5819
4	1029	9800	34944	55296	32768											133837
5	7203	85064	401408	946176	1114112	524288										3078251
6	50421	710696	4170880	13045760	22937600	21495808	8388608									70799773
7	352947	5781608	40567296	158054400	369295360	517472256	402653184	134217728								1628394779
8	2470629	46118408	376476800	1755457536	5113937920	9531031552	11098128384	7381975040	2147483648							37453079917
9	17294403	362358920	3373232128	18311831552	63884886016	148540227584	230183403520	229243879424	133143986176	34359738368						861420838091
10	121060821	2813222888	29410367616	182154534912	740183506944	2061939769344	3987927465984	5287641612288	4599909974016	2370821947392	549755813888					19812679276093
11	847425747	21629533352	250884139520	1745647626240	8095757107200	26276514496512	60906528571392	100820330741760	116801635614720	90194313216000	41781441855488	8796093022208				455691623350139
12	5931980229	164965545416	2102261510272	16233679616000	84600661770240	313467715190784	846769931943936	1680246772334592	2430736741171200	2500186362347520	1735579104444416	730075720843264	140737488355328			10480907337053197
13	41523861603	1249670501576	17355279298560	147271941476352	851943506247680	3547884594659328	10942872966660096	25310046317445120	43899105545551872	56393092395171840	52152035528671232	32879795717013504	12666373951979520	2251799813685248		241060868752223531
14	290667031221	9412075296680	141481683115136	1308588059111424	8319955607355392	38466288262578176	133366264281169920	352256291688677376	712254479897985024	1097137335495032832	1267353727023448064	1064591138477834240	614741349136072704	218424581927469056	36028797018963968	5544399981301141213

Technical details

Node id: n-f212557825a249a7 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-74fee7c1a09463d3

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:        3,         8;
  2:       21,       104,        128;
  3:      147,      1064,       2560,        2048;
  4:     1029,      9800,      34944,       55296,       32768;
  5:     7203,     85064,     401408,      946176,     1114112,       524288;
  6:    50421,    710696,    4170880,    13045760,    22937600,     21495808,      8388608;
  7:   352947,   5781608,   40567296,   158054400,   369295360,    517472256,    402653184,    134217728;
  8:  2470629,  46118408,  376476800,  1755457536,  5113937920,   9531031552,  11098128384,   7381975040,   2147483648;
  9: 17294403, 362358920, 3373232128, 18311831552, 63884886016, 148540227584, 230183403520, 229243879424, 133143986176, 34359738368;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(16*n-16)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (16*n - 16)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (16*n - 16)*x)*v[i - 1])/(n - 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Mirror of the triangle A193728.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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.