A119308 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-8\,x^{2} + 4\,n x^{2}\right)\,P_{n-3}(x) + \left(2 + 2\,x - 4\,n - 4\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(7 + 6\,x + 5\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

\(P_{2}(x) = 9 + 14\,x + 4\,x^{2}\)

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	2														5
2	9	14	4													27
3	28	70	48	8												154
4	90	312	352	144	16											914
5	297	1320	2100	1440	400	32										5589
6	1001	5434	11220	11000	5200	1056	64									34975
7	3432	22022	56056	72072	49280	17248	2688	128								222926
8	11934	88400	267904	428064	387296	198016	53760	6656	256							1442286
9	41990	352716	1241136	2376192	2690688	1834560	733824	159744	16128	512						9447490
10	149226	1401820	5620200	12558240	17136000	14714112	7902720	2553600	456960	38400	1024					62532302
11	534888	5556892	25013120	63954000	102326400	106419456	72382464	31595520	8448000	1267200	90112	2048				417590100
12	1931540	21989840	109830336	316415968	581555040	712191744	589400064	327444480	118914048	26808320	3424256	208896	4096			2810118628
13	7020405	86919300	477090380	1529779680	3178542224	4487353728	4389802560	2979744768	1383452928	425677824	82148352	9052160	479232	8192		19037071733
14	25662825	343311570	2054363220	7258272840	16832575760	26946119200	30487609152	24548464512	13978508544	5521291776	1460659200	244328448	23482368	1089536	16384	129725755335

Technical details

Node id: n-eaddbafbdfb5ca13 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-a10e516e57874c56

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     3,      2;
  2:     9,     14,       4;
  3:    28,     70,      48,       8;
  4:    90,    312,     352,     144,      16;
  5:   297,   1320,    2100,    1440,     400,      32;
  6:  1001,   5434,   11220,   11000,    5200,    1056,     64;
  7:  3432,  22022,   56056,   72072,   49280,   17248,   2688,    128;
  8: 11934,  88400,  267904,  428064,  387296,  198016,  53760,   6656,   256;
  9: 41990, 352716, 1241136, 2376192, 2690688, 1834560, 733824, 159744, 16128, 512;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = (4*n-8)*x^2*P(n-3,x) - (4*n-2+(4*n-2)*x+4*n*x^2)*P(n-2,x) + (5*n+7+(4*n+6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 9 + 14*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 9 + 14*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 8)*x^2*P[n - 3, x] - (4*n - 2 + (4*n - 2)*x + 4*n*x^2)*P[n - 2, x] + (5*n + 7 + (4*n + 6)*x)*P[n - 1, x])/(n + 3)]];
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 + 2*x; v[3] = 9 + 14*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8)*x^2*v[i - 3] - (4*n - 2 + (4*n - 2)*x + 4*n*x^2)*v[i - 2] + (5*n + 7 + (4*n + 6)*x)*v[i - 1])/(n + 3)); 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle for first differences of Catalan numbers.?
  - 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

References

[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.