Real-rooted atlas

A039755 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
366113
4243816179
5116256184441601
66481844189682412615339
740881429619236122643606370153861
828640119160199088168804749361579811001607527
9219920106316821289602265696136125644816070176328817560625
1018322241010715223642944303806082316356810512872267003231798498501102637235
1116430176101954912273253264412380480382473456222758864794839441598875214717022953411506225085
1215755404810872090563288406144570932264062529592964463865328206753708859587575296672632694501488584123726435583
1316068790401221429964841189952640809679140481024435059688683940569649687988736188126428484464490460591596576333077042657321398852249097
1417350255744144126757696536540555136117899175436816944313696321666320170800113859006304053688672051216950339020033598294300366902207216175532079717417120170905995
15197553645440178133686924872600295189121764884988096028428801569312318616766329282536538906859214366796702400569674450845615223704963602547977667962307593720879283309423914981134408217821205
Technical details
Node id
n-29b95372502618c2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-078627b5c58f7a3e

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 -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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       38,       16,        1;
   5:     116,      256,      184,       44,        1;
   6:     648,     1844,     1896,      824,      126,        1;
   7:    4088,    14296,    19236,    12264,     3606,      370,       1;
   8:   28640,   119160,   199088,   168804,    74936,    15798,    1100,      1;
   9:  219920,  1063168,  2128960,  2265696,  1361256,   448160,   70176,   3288,    1;
  10: 1832224, 10107152, 23642944, 30380608, 23163568, 10512872, 2670032, 317984, 9850, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-2+(2*n-3)*x)*P(n-1,x) - (2+4*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 2 + (2*n - 3)*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 2 + (2*n - 3)*x)*v[i - 1] - (2 + 4*x + 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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