A136157 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 3\)

\(P_{1}(x) = 4 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	3															3
1	4	12														16
2	4	28	48													80
3	4	44	160	192												400
4	4	60	336	832	768											2000
5	4	76	576	2176	4096	3072										10000
6	4	92	880	4480	12800	19456	12288									50000
7	4	108	1248	8000	30720	70656	90112	49152								250000
8	4	124	1680	12992	62720	193536	372736	409600	196608							1250000
9	4	140	2176	19712	114688	444416	1146880	1900544	1835008	786432						6250000
10	4	156	2736	28416	193536	903168	2924544	6488064	9437184	8126464	3145728					31250000
11	4	172	3360	39360	307200	1677312	6537216	18186240	35389440	45875200	35651584	12582912				156250000
12	4	188	4048	52800	464640	2906112	13246464	44335104	108134400	187432960	219152384	155189248	50331648			781250000
13	4	204	4800	68992	675840	4764672	24870912	97320960	285474816	619970560	968884224	1031798784	671088640	201326592		3906250000
14	4	220	5616	88192	951808	7468032	43929600	196804608	674758656	1761869824	3448766464	4907335680	4798283776	2885681152	805306368	19531250000

Technical details

Node id: n-e3e112b1337439e7 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e75183813a6a02ba

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: 3;
  1: 4,  12;
  2: 4,  28,   48;
  3: 4,  44,  160,   192;
  4: 4,  60,  336,   832,    768;
  5: 4,  76,  576,  2176,   4096,   3072;
  6: 4,  92,  880,  4480,  12800,  19456,   12288;
  7: 4, 108, 1248,  8000,  30720,  70656,   90112,   49152;
  8: 4, 124, 1680, 12992,  62720, 193536,  372736,  409600,  196608;
  9: 4, 140, 2176, 19712, 114688, 444416, 1146880, 1900544, 1835008, 786432;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 4 + 12*x.

%t Mathematica

P[0, x_] := 3;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (4*n - 4)*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] = 3; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (4*n - 4)*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 by columns, (3, 1, 0, 0, 0, ...) in every column.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.