Real-rooted atlas

A008957 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(1 + 2\,x + x^{2} - 2\,n - 4\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
277115
33759241121
42645834088911345
52433690269072782346119371
62791397298128065771621985313711351663
7386906160456126328222139777870413151269546817791217
8634611930572553602555086193862134851583101449251224742218531205355905
9121159373665995685153599805519162904941385580997573452597123280083103671208739016332211795
102655174768164194706594345323288863931140050567420808463072748762797072758541569561960902454753495351225296019663
1166028903633454104255980135789298087423056776965602432653487145163645197947468999146585217050721271020971137045798227935139811219135078745321
1218455791009931398283040957246664580166326899316848107601101568180644158882023626335447103128123964158657591410623116577661770293288036755712113645255924171417787608507841
13575068472621624761910987626731755811454178804379018659207884652938878760634004986471734437299319747614704317813726885379266083759408239983775928704508684042656479840276381060731422369634121369777637799163
14198331215241135117828838938336475720232830496716191722862463941077942705475758631160812919992910644638332201709013033564411153641115191303364085686125173028392129345394067921172892527170079639766676681985730651360738947849911213510798425183855
Technical details
Node id
n-d7dc1caea8809a46 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4c47ff87ed3f3de6

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:         2,         1;
  2:         7,         7,          1;
  3:        37,        59,         24,          1;
  4:       264,       583,        408,         89,          1;
  5:      2433,      6902,       6907,       2782,        346,         1;
  6:     27913,     97298,     128065,      77162,      19853,      1371,         1;
  7:    386906,   1604561,    2632822,    2139777,     870413,    151269,      5468,        1;
  8:   6346119,  30572553,   60255508,   61938621,   34851583,  10144925,   1224742,    21853,     1;
  9: 121159373, 665995685, 1535998055, 1916290494, 1385580997, 573452597, 123280083, 10367120, 87390, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^2+1+n^2*x)*P(n-1,x) - (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*d/dx P(n-1,x) + (1+3*x+3*x^2+x^3)*d^2/dx^2 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[(n^2 + 1 + n^2*x)*P[n - 1, x] - (2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)*D[P[n - 1, x], x] + (1 + 3*x + 3*x^2 + x^3)*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; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + 1 + n^2*x)*v[i - 1] - (2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)*deriv(v[i - 1], x) + (1 + 3*x + 3*x^2 + x^3)*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)

polytool c48629374597

Properties

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.