A380114 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4 - 8\,x + 4\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	8	48	64													120
3	32	256	640	512												1440
4	128	1280	4608	7168	4096											17280
5	512	6144	28672	65536	73728	32768										207360
6	2048	28672	163840	491520	819200	720896	262144									2488320
7	8192	131072	884736	3276800	7208960	9437184	6815744	2097152								29859840
8	32768	589824	4587520	20185088	55050240	95420416	102760448	62914560	16777216							358318080
9	131072	2621440	23068672	117440512	381681664	822083584	1174405120	1073741824	570425344	134217728						4299816960
10	524288	11534336	113246208	654311424	2466250752	6341787648	11274289152	13690208256	10871635968	5100273664	1073741824					51597803520
11	2097152	50331648	545259520	3523215360	15099494400	45097156608	95831457792	144955146240	153008209920	107374182400	45097156608	8589934592				619173642240
12	8388608	218103808	2583691264	18454937600	88583700480	301184581632	744103084032	1346472247296	1771674009600	1653562408960	1039382085632	395136991232	68719476736			7430083706880
13	33554432	939524096	12079595520	94489280512	501974302720	1913407930368	5385888989184	11338713661440	17858474016768	20787641712640	17386027614208	9895604649984	3435973836800	549755813888		89161004482560
14	134217728	4026531840	55834574848	474593886208	2763811454976	11669426143232	36850819399680	88441966559232	162143605358592	226018358984704	235845244157952	178670639513600	92908732547072	29686813949952	4398046511104	1069932053790720

Technical details

Node id: n-acacc82d3699f634 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e0f7dc6dd641b7f6

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:      2,       8;
  2:      8,      48,       64;
  3:     32,     256,      640,       512;
  4:    128,    1280,     4608,      7168,      4096;
  5:    512,    6144,    28672,     65536,     73728,     32768;
  6:   2048,   28672,   163840,    491520,    819200,    720896,     262144;
  7:   8192,  131072,   884736,   3276800,   7208960,   9437184,    6815744,    2097152;
  8:  32768,  589824,  4587520,  20185088,  55050240,  95420416,  102760448,   62914560,  16777216;
  9: 131072, 2621440, 23068672, 117440512, 381681664, 822083584, 1174405120, 1073741824, 570425344, 134217728;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(8*n-8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (8*n - 8)*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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (8*n - 8)*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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is defined in A357…?
  - 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 ?

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.