Real-rooted atlas

A299989 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(8 + 7\,x + x^{2}\right)\,P_{n-1}(x) + \left(7\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0112
182210141
2643864461672311087
351262381424211137372157540136466
44096971543984625526953376311031771669614346111511407
532768148598210441570238245612378978512117311342732056236754130298086175738861
62621442251910626400428695020288714620705431144061929503969672132910724217326272233497143078549711514504116106
72097152339621598653903696236183667297827960278819495716255961228697096238776473735903010116952151216101607156716851832825193171481312888232641

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-8e135950a7b2bef7 · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-fcb14b25d1354b2a

Observed on the stored rows ?

  • Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1,           1;
  1:         8,          22,            10,              1;
  2:        64,         386,           446,            167,              23,               1;
  3:       512,        6238,         14242,          11137,            3721,             575,              40,               1;
  4:      4096,       97154,        398462,         552695,          337631,          103177,           16696,            1434,              61,              1;
  5:     32768,     1485982,      10441570,       23824561,        23789785,        12117311,         3427320,          562367,           54130,           2980,             86,             1;
  6:    262144,    22519106,     264004286,      950202887,      1462070543,      1144061929,       503969672,       132910724,        21732627,        2233497,         143078,          5497,          115,           1;
  7:   2097152,   339621598,    6539036962,    36183667297,     82796027881,     94957162559,     61228697096,     23877647373,      5903010116,      952151216,      101607156,       7168518,       328251,        9317,        148,        1;
  8:  16777216,  5109004034,  159995578622,  1338697383575,   4451136050111,   7260425326729,   6595363559064,   3622189497310,   1271020505849,   295804503929,    46776792724,    5093405438,    383022402,    19720134,     678336,    14820,     185,     1;
  9: 134217728, 76752501022, 3885891700450, 48619468794001, 231217591970425, 525512533719167, 655507451329624, 491169093182635, 235183803704454, 75155705193684, 16524634281706, 2551991967887, 280282568474, 21995734070, 1229147592, 48248754, 1293276, 22434, 226, 1;
%F Formula
The row polynomials satisfy P(n,x) = (8+7*x+x^2)*P(n-1,x) + (7*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1 + x.
%t Mathematica
P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(8 + 7*x + x^2)*P[n - 1, x] + (7*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 7*x + x^2)*v[i - 1] + (7*x + 2*x^2)*deriv(v[i - 1], x) + x^2*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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-cfdeae139dd131be
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-cfdeae139dd131be
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-cfdeae139dd131be

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741