Real-rooted atlas

A178820 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 9\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x) + \left(9\,x + 3\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1201232
225039090730
325007740432054015100
421875121485123795359102835305900
5175000166000827619201398600249480136086258616
6131250020707596530417164179924012655440153090061236131108628
7937500024228324092225952010633064404864860009797760085730402624402830523280
8644531252703787515149485936052427296949015757749315470113875067527999044746020108256563169800375
94296875002909821266023006118960051288980490045426752910018839469726039513141360425231532022084326043302601459131751220
10279296875030429719291434033580406541023183308862012039582632220666990549225019161822459782978295091202490982065010414275301688801434891749306700
111777343750031089627539884880887457846419530730209790829955978692640021574872803484081558792170856171717814227962056566051540137524727340472864392064481508863480885327060
12111083984375311562779305056827806357813353600911066382630709804785505315565170674018450003153873005042760863928366881580138944699703810132131117868757225958990252079528633024180565522100777137383035
13683593750000307179662027280935883794139840064567503799016400161808672904678800186628297356380880113341884703407360392907772972915208149146328683360103341990928176079914082248000364062594168088984481040892820880584570048081407360
14415039062500029865212369257201261263469860637201131740474226944400357540351165728640051218473870934370003845337148871978760165335426323404480043009705587162600069841920958009200716094959363784045910446806952017692436165040371859896520325241892015964376902137128840
Technical details
Node id
n-3bc20c394015b74b · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

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:        20,          12;
  2:       250,         390,           90;
  3:      2500,        7740,         4320,          540;
  4:     21875,      121485,       123795,        35910,         2835;
  5:    175000,     1660008,      2761920,      1398600,       249480,        13608;
  6:   1312500,    20707596,     53041716,     41799240,     12655440,      1530900,       61236;
  7:   9375000,   242283240,    922259520,   1063306440,    486486000,     97977600,     8573040,     262440;
  8:  64453125,  2703787515,  14948593605,  24272969490,  15757749315,   4701138750,   675279990,   44746020,   1082565;
  9: 429687500, 29098212660, 230061189600, 512889804900, 454267529100, 188394697260, 39513141360, 4252315320, 220843260, 4330260;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+15+(3*n+9)*x)*P(n-1,x) + (3*n+9)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 15 + (3*n + 9)*x)*P[n - 1, x] + (3*n + 9)*x*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n + 15 + (3*n + 9)*x)*v[i - 1] + (3*n + 9)*x*deriv(v[i - 1], x))/(n)); 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 Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.