Real-rooted atlas

A038301 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(18 + 22\,x + 9\,n + 11\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 198 + 242\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0198242440
126736534399313200
23207611761214374858564352000
33608551764180323433026353808052558800000
438972342381643058217940711552604348377010629366211200000
540920957300087018916932555149426046013697387556696500581364101974928000000
642090127236010442161320382879226896688280328737301202410740208898215341841714871048112640000000
7426162537941669225928178251688692435726350136665693034930650899856376397772134452138904554888212215292192534400000000

All 15 rows

Technical details
Node id
n-6a03653d7a50bc7a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6bae7ffb6b8d3725

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          198,           242;
  1:         2673,          6534,           3993;
  2:        32076,        117612,         143748,          58564;
  3:       360855,       1764180,        3234330,        2635380,          805255;
  4:      3897234,      23816430,       58217940,       71155260,        43483770,        10629366;
  5:     40920957,     300087018,      916932555,     1494260460,      1369738755,       669650058,       136410197;
  6:    420901272,    3601044216,    13203828792,    26896688280,     32873730120,     24107402088,      9821534184,      1714871048;
  7:   4261625379,   41669225928,   178251688692,   435726350136,    665693034930,    650899856376,    397772134452,    138904554888,    21221529219;
  8:  42616253790,  468778791690,  2291807426040,  6535895252040,  11982474628740,  14645246768460,  11933164033560,   6250704969960,  1909937629710,   259374246010;
  9: 421900912521, 5156566708590, 28361116897245, 92436232850280, 197710831374210, 289975886015508, 295345809830610, 206273264008680, 94541912670645, 25678050354990, 3138428376721;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (9*n+18+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 198 + 242*x.
%t Mathematica
P[0, x_] := 198 + 242*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n + 18 + (11*n + 22)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 198 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((9*n + 18 + (11*n + 22)*x)*v[i - 1])/(n + 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. [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.