A193726 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-7 - 10\,x + 7\,n + 10\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

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	4														7
2	21	58	40													119
3	147	616	860	400												2023
4	1029	5782	12180	11400	4000											34391
5	7203	50764	143080	201600	142000	40000										584647
6	50421	427378	1509200	2842000	3010000	1700000	400000									9938999
7	352947	3495856	14838180	34986000	49490000	42000000	19800000	4000000								168962983
8	2470629	28000462	138825820	393283800	696290000	788900000	558600000	226000000	40000000							2872370711
9	17294403	220709524	1251785360	4141244800	8806868000	12485200000	11799200000	7168000000	2540000000	400000000						48830302087
10	121060821	1717910698	10969592760	41506567200	103060524000	175465080000	207446400000	168168000000	89460000000	28200000000	4000000000					830115135479
11	847425747	13235983096	93966256300	400241898000	1136489340000	2258860800000	3206775600000	3251640000000	2307900000000	1092000000000	310000000000	40000000000				14111957303143
12	5931980229	101126139142	790123625060	3741355849000	11957844360000	27176919000000	45036037200000	54829236000000	48671700000000	30723000000000	13090000000000	3380000000000	400000000000			239903274153431
13	41523861603	767202776284	6542126766840	34090727193600	121118469010000	309816876600000	587021450400000	834165024000000	888994260000000	701778000000000	398860000000000	154560000000000	36600000000000	4000000000000		4078355660608327
14	290667031221	5785658050018	53466915130720	304056358023600	1188736555006000	3379902826300000	7207318918800000	11709369672000000	14564610060000000	13802388600000000	9809800000000000	5070520000000000	1801800000000000	394000000000000	40000000000000	69332046230341559

Technical details

Node id: n-e2d2c5e8ff2d490b · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8296d10b798c08ec

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,         4;
  2:       21,        58,         40;
  3:      147,       616,        860,        400;
  4:     1029,      5782,      12180,      11400,       4000;
  5:     7203,     50764,     143080,     201600,     142000,       40000;
  6:    50421,    427378,    1509200,    2842000,    3010000,     1700000,      400000;
  7:   352947,   3495856,   14838180,   34986000,   49490000,    42000000,    19800000,    4000000;
  8:  2470629,  28000462,  138825820,  393283800,  696290000,   788900000,   558600000,  226000000,   40000000;
  9: 17294403, 220709524, 1251785360, 4141244800, 8806868000, 12485200000, 11799200000, 7168000000, 2540000000, 400000000;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(10*n-10)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (10*n - 10)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (10*n - 10)*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 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.?
  - 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

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.