Real-rooted atlas

A253283 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-9 + 9\,n - 2\,n^{2}\right)\,P_{n-2}(x) + \left(46 + 30\,x - 48\,n - 32\,n x + 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
358316
425574210134
5129384420200351168
668125053630259091012610442
7365316008288752744014490403246294960
81982510084921764425830018207076230175561716874190
910854562873615795362247840198198011088003843847550464358121760
10598417388865711145780184938601957725013711698635434818790203217502431075995090
113317445239000407696111514586792018021003015163948888017930348004808970390136136092378715138576
121847463314614647352232103011132507101572220650154488133810777646885338476001840023904203199057274363527166761021654
13103274625889928064349540448482747064401314949135514777642880120431351047171411104309697245094595072019399380023984688135207864167247792
145791688255399971161231198900666018633621010628620002013445914482012536573649687226263696452906239501733334603047528481008842590121000537725200300610989042458
153256957317326682365521514238493954299096110408352198855001174607546112123316814821298183033520059574526011027415203816094189809714234288075843988630100416024000200583005834025197296

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 - n\right)\,P_{n}(x) = \left(4 - n\right)\,P_{n-3}(x) + \left(-23 - 14\,x + 7\,n + 4\,n x\right)\,P_{n-2}(x) + \left(12 + 6\,x - 7\,n - 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)
\(P_{3}(x) = 5 + 8\,x + 3\,x^{2}\)
Technical details
Node id
n-d6d09a63aa4e7864 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-911e771a8b01ca42)
Fingerprint
f6-4930f908745e0f57

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:      1,       1;
   3:      5,       8,        3;
   4:     25,      57,       42,       10;
   5:    129,     384,      420,      200,       35;
   6:    681,    2505,     3630,     2590,      910,      126;
   7:   3653,   16008,    28875,    27440,    14490,     4032,     462;
   8:  19825,  100849,   217644,   258300,   182070,    76230,   17556,    1716;
   9: 108545,  628736,  1579536,  2247840,  1981980,  1108800,  384384,   75504,   6435;
  10: 598417, 3888657, 11145780, 18493860, 19577250, 13711698, 6354348, 1879020, 321750, 24310;
%F Formula
The row polynomials satisfy (2*n^2-7*n+5)*P(n,x) = -(2*n^2-9*n+9)*P(n-2,x) + (12*n^2-48*n+46+(8*n^2-32*n+30)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 9)*P[n - 2, x] + (12*n^2 - 48*n + 46 + (8*n^2 - 32*n + 30)*x)*P[n - 1, x])/(2*n^2 - 7*n + 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 9*n + 9)*v[i - 2] + (12*n^2 - 48*n + 46 + (8*n^2 - 32*n + 30)*x)*v[i - 1])/(2*n^2 - 7*n + 5)); 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

References

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