A038321 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(18 + 7\,x + 7\,n\right)\,P_{n}(x) = \left(275 + 252\,x + 49\,x^{2} + 77\,n + 49\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 275 + 252\,x + 49\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	275	252	49								576
1	3872	5775	2646	343							12636
2	51909	108416	80850	24696	2401						268272
3	673486	1816815	1897280	943250	216090	16807					5563728
4	8535703	28286412	38153115	26561920	9904125	1815156	117649				113374080
5	106293660	418249447	693017094	623167545	325383520	97060425	14823774	823543			2278819008
6	1305640457	5952444960	11710984516	12936319088	8724345630	3644295424	905897300	118590192	5764801		45304282368
7	15862557194	82255348791	187502016240	245930674836	203747025636	109926754938	38265101952	8153075700	933897762	40353607	892616806656

All 15 rows

Technical details

Node id: n-aeb3908f9ea8b1da · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1d0954627bd2ed55

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 2)
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:           275,            252,             49;
  1:          3872,           5775,           2646,            343;
  2:         51909,         108416,          80850,          24696,           2401;
  3:        673486,        1816815,        1897280,         943250,         216090,          16807;
  4:       8535703,       28286412,       38153115,       26561920,        9904125,        1815156,         117649;
  5:     106293660,      418249447,      693017094,      623167545,      325383520,       97060425,       14823774,         823543;
  6:    1305640457,     5952444960,    11710984516,    12936319088,     8724345630,     3644295424,      905897300,      118590192,       5764801;
  7:   15862557194,    82255348791,   187502016240,   245930674836,   203747025636,   109926754938,    38265101952,     8153075700,     933897762,     40353607;
  8:  190993762971,  1110379003580,  2878937207685,  4375047045600,  4303786809630,  2852458358904,  1282478807610,   382651019520,   71339412375,   7263649260,   282475249;
  9: 2282493364888, 14706519748767, 42749591637830, 73892721663915, 84219655627800, 66278316868302, 36606548939268, 14107266883710, 3683016062880, 610348305875, 55930099302, 1977326743;

%F Formula

The row polynomials satisfy (7*n+18+7*x)*P(n,x) = (77*n+275+(49*n+252)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 275 + 252*x + 49*x^2.

%t Mathematica

P[0, x_] := 275 + 252*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((77*n + 275 + (49*n + 252)*x + 49*x^2)*P[n - 1, x])/(7*n + 18 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 275 + 252*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((77*n + 275 + (49*n + 252)*x + 49*x^2)*v[i - 1])/(7*n + 18 + 7*x)); 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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - 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
  - Parent-row provers of polytool polytool ? 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.