Real-rooted atlas

A113953 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 12\,x\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14610
2246036120
31284805762161400
4704345661924752129616400
53840234245587264800362887776192000
620992152832455040706752601344264384466562248000
71146889707523465216674611277137925163264186624027993626320000
862668860456962515968058890240846236167626700841990400128770561679616308160000
934242563708518417628364848207052883449958494668134470245273632808499287340032100776963608000000
1018710528224772096120147148837590220807614093312104193423369741026304613395763224858316805845063686046617642243200000
111022361601349124096800980992028216885248654713487361049493749761184599572489403610112051382812672183817175043869835264362797056494592000000
12558628864803261644852443021312205416235008537207717888987349450752130649639731212528681246728632420024324163297771521332674519042539579392021767823365790784000000
13305240473647502065664338235752448145817586892842437016944648797895294976133571544514561502425917849612509672472576760841861529632825877626889505282867201654354575361306069401667799680000000
1416678649856279287169024215389097164810134764126208324865185546247499907956736012851615416320016594086518784016212459847680011918125154304064842959683584253000154972166687075336192107097690931278364164096793812480000000
Technical details
Node id
n-587005f11f3bbeb8 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-87f1729e0e693bae

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:       4,        6;
  2:      24,       60,        36;
  3:     128,      480,       576,       216;
  4:     704,     3456,      6192,      4752,      1296;
  5:    3840,    23424,     55872,     64800,     36288,      7776;
  6:   20992,   152832,    455040,    706752,    601344,    264384,     46656;
  7:  114688,   970752,   3465216,   6746112,   7713792,   5163264,   1866240,    279936;
  8:  626688,  6045696,  25159680,  58890240,  84623616,  76267008,  41990400,  12877056,  1679616;
  9: 3424256, 37085184, 176283648, 482070528, 834499584, 946681344, 702452736, 328084992, 87340032, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (8+12*x)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 6*x;
P[n_, x_] := P[n, x] = Expand[(8 + 12*x)*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 12*x)*v[i - 2] + (4 + 6*x)*v[i - 1]); 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. [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
  2. [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