Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

A088729 (tail) ↗ nonnegative coefficients

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

Recurrence from A088729

\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(4 + 3\,x\right)\,P_{n-1}'(x) + \left(2\,x\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 13 + 9\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	13	9	1															23
2	75	79	18	1														173
3	541	765	265	30	1													1602
4	4683	8311	3870	665	45	1												17575
5	47293	100989	59101	13650	1400	63	1											222497
6	545835	1362439	960498	278901	38430	2618	84	1										3188806
7	7087261	20246445	16700545	5844510	1012431	92610	4494	108	1									50988405
8	102247563	328972471	310819590	127331905	26587575	3056907	199080	7230	135	1								899222457
9	1622632573	5805917469	6182314501	2902305450	709559950	98334621	8047347	392040	11055	165	1							17329515172
10	28091567595	110645911399	131102078778	69392377901	19445781960	3155560276	311237388	19048227	720225	16225	198	1						362164300173
11	526858348381	2265191981325	2955981690025	1741641891990	550412723861	102389807520	11824946848	872107236	41414802	1250535	23023	234	1					8155216185781
12	10641342970443	49589790516631	70668432599310	45872193348345	16141221201105	3386724794453	448389701760	38719252000	2215403190	83995912	2072070	31759	273	1				196789115887252
13	230283190977853	1156201277261949	1786538324270701	1266783600823650	491193999032100	114767002449099	17139464755413	1696669809120	113654402290	5191135950	160758598	3300570	42770	315	1			5064722539020379
14	5315654681981355	28605950745797959	47637852615465858	36635701495438501	15520713733251090	3996715082570574	664722780934212	74225774104053	5699743141950	304755372350	11367520164	292923358	5083260	56420	360	1		138457553073641465
15	130370767029135901	748605590542359405	1336603460175514705	1108094819890771470	509249312448439291	143298298817995230	26269180181842962	3266904371872284	282974805959703	17343258748530	757215304274	23498738172	511707378	7604100	73100	408	1	4006059432756066914

Technical details

Node id: n-4f94f81718ae3273 · revision 0 · JSON
Definition source: oeis_catalog:A088729
Fingerprint: f6-ad16784f8d87fb89

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 1)
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 ?

Row sums ? A075729

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:          13,            9,            1;
   2:          75,           79,           18,           1;
   3:         541,          765,          265,          30,           1;
   4:        4683,         8311,         3870,         665,          45,          1;
   5:       47293,       100989,        59101,       13650,        1400,         63,         1;
   6:      545835,      1362439,       960498,      278901,       38430,       2618,        84,        1;
   7:     7087261,     20246445,     16700545,     5844510,     1012431,      92610,      4494,      108,      1;
   8:   102247563,    328972471,    310819590,   127331905,    26587575,    3056907,    199080,     7230,    135,     1;
   9:  1622632573,   5805917469,   6182314501,  2902305450,   709559950,   98334621,   8047347,   392040,  11055,   165,   1;
  10: 28091567595, 110645911399, 131102078778, 69392377901, 19445781960, 3155560276, 311237388, 19048227, 720225, 16225, 198, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (4+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 13 + 9*x + x^2.

%t Mathematica

P[1, x_] := 13 + 9*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + (4 + 3*x)*D[P[n - 1, x], x] + 2*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 13 + 9*x + x^2; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + (4 + 3*x)*deriv(v[i - 1], x) + 2*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Iterated real-rootedness preservers

Proof chains ? JSON

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.