A038305 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(25 + 24\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 625 + 1200\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	625	1200	576								2401
1	15625	45000	43200	13824							117649
2	390625	1500000	2160000	1382400	331776						5764801
3	9765625	46875000	90000000	86400000	41472000	7962624					282475249
4	244140625	1406250000	3375000000	4320000000	3110400000	1194393600	191102976				13841287201
5	6103515625	41015625000	118125000000	189000000000	181440000000	104509440000	33443020800	4586471424			678223072849
6	152587890625	1171875000000	3937500000000	7560000000000	9072000000000	6967296000000	3344302080000	917294284800	110075314176		33232930569601
7	3814697265625	32958984375000	126562500000000	283500000000000	408240000000000	391910400000000	250822656000000	103195607040000	24766945689600	2641807540224	1628413597910449

All 15 rows

Technical details

Node id: n-245f1aee5991880d · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-49793563bcf540fa

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:              625,              1200,                576;
  1:            15625,             45000,              43200,              13824;
  2:           390625,           1500000,            2160000,            1382400,             331776;
  3:          9765625,          46875000,           90000000,           86400000,           41472000,            7962624;
  4:        244140625,        1406250000,         3375000000,         4320000000,         3110400000,         1194393600,          191102976;
  5:       6103515625,       41015625000,       118125000000,       189000000000,       181440000000,       104509440000,        33443020800,         4586471424;
  6:     152587890625,     1171875000000,      3937500000000,      7560000000000,      9072000000000,      6967296000000,      3344302080000,       917294284800,       110075314176;
  7:    3814697265625,    32958984375000,    126562500000000,    283500000000000,    408240000000000,    391910400000000,    250822656000000,    103195607040000,     24766945689600,     2641807540224;
  8:   95367431640625,   915527343750000,   3955078125000000,  10125000000000000,  17010000000000000,  19595520000000000,  15676416000000000,   8599633920000000,   3095868211200000,   660451885056000,    63403380965376;
  9: 2384185791015625, 25177001953125000, 120849609375000000, 348046875000000000, 668250000000000000, 898128000000000000, 862202880000000000, 591224832000000000, 283787919360000000, 90812134195200000, 17435929765478400, 1521681143169024;

%F Formula

The row polynomials satisfy P(n,x) = (25+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 1200*x + 576*x^2.

%t Mathematica

P[0, x_] := 625 + 1200*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(25 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 625 + 1200*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 24*x)*v[i - 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.