Real-rooted atlas

A034870 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(36 + 64\,x + 16\,x^{2}\right)\,P_{n-1}(x) + \left(64\,x + 32\,x^{2}\right)\,P_{n-1}'(x) + \left(16\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1366416116
2129687049856307225623184
346656953344278809623961607731209830440967059776
41679616983203846387343361148485632802185216253231104380764162621440655362983399680
56046617699395338241346575605764554404659205822350950403435795578881038275051521682964480014588313606291456010485761648092623872
6217678233699782321766427348044578816164957911252992356826781515776351335966834688179722337124352515321836339208614730268672847081635840477060136961409286144167772161142234168922112
7783641640969992163583590454560425948610565696287812354048019953499676737536030865318221132595224615858848713932811120149805400064030156496788520960507548062973952053695434509516835395630792704139848161689630064771072268435456963872942447214592

All 15 rows

Technical details
Node id
n-2137bba207333614 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-14e66869ea533ba5

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:              36,                 64,                    16;
  2:            1296,               8704,                  9856,                   3072,                     256;
  3:           46656,             953344,               2788096,                2396160,                  773120,                    98304,                     4096;
  4:         1679616,           98320384,             638734336,             1148485632,               802185216,                253231104,                 38076416,                  2621440,                    65536;
  5:        60466176,         9939533824,          134657560576,           455440465920,            582235095040,             343579557888,             103827505152,              16829644800,               1458831360,                62914560,                 1048576;
  6:      2176782336,       997823217664,        27348044578816,        164957911252992,         356826781515776,          351335966834688,          179722337124352,           51532183633920,            8614730268672,            847081635840,             47706013696,             1409286144,              16777216;
  7:     78364164096,     99921635835904,      5456042594861056,      56962878123540480,      199534996767375360,       308653182211325952,       246158588487139328,       111201498054000640,        30156496788520960,        5075480629739520,         536954345095168,         35395630792704,         1398481616896,          30064771072,           268435456;
  8:   2821109907456,   9997178890092544,   1078978079459639296,   19155944710342705152,   105776295616693927936,    247871676604169060352,    293873602402040414208,    196503981318513623040,     79510550217970679808,    20382450772051230720,     3405268524506021888,     375887235574136832,     27393824326156288,     1293987746938880,      37838661877760,      618475290624,      4294967296;
  9: 101559956668416, 999898440043331584, 212439477885296705536, 6344795235184111779840, 54277941881406048174080, 188176797258189002244096, 321701918451037345153024, 306356191444790799237120, 176108358772914231705600, 64542082932745206497280, 15652339673104306405376, 2573460414698474176512, 290733778126009008128, 22639603848006598656, 1204867124007272448, 42771277198393344, 962914487894016, 12369505812480, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (36+64*x+16*x^2)*P(n-1,x) + (64*x+32*x^2)*d/dx P(n-1,x) + 16*x^2*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[(36 + 64*x + 16*x^2)*P[n - 1, x] + (64*x + 32*x^2)*D[P[n - 1, x], x] + 16*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; for(n = 1, N, my(i = n + 1); v[i] = (36 + 64*x + 16*x^2)*v[i - 1] + (64*x + 32*x^2)*deriv(v[i - 1], x) + 16*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)

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