Real-rooted atlas

A038321 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 57\,x + 39\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 1404\,x + 1521\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0324140415213249
15832379088213459319185193
210497690979229568244270968231344110556001
31889568204703208870472019219356020820969090224199601692057
43401222444215891223950274406918968160112433232609744213492351874376134296447249
56122200329285337152603546914882179474970404722195769206138854499964433617138861372310066791954897493193
61101996057619101264998414485125957126276887914752169999047691202946650159980831922043399792197612649617765352009260481111429157112001
71983592903683868006162176335227200721921694759736983045507969145194881193393314792224172379034358876816006624619038568670255001979222087283611587596351461955384057

All 15 rows

Technical details
Node id
n-85b7a886c31a0704 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a285c8a8f5f97b30

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            324,             1404,              1521;
  1:           5832,            37908,             82134,              59319;
  2:         104976,           909792,           2956824,            4270968,            2313441;
  3:        1889568,         20470320,          88704720,          192193560,          208209690,            90224199;
  4:       34012224,        442158912,        2395027440,         6918968160,        11243323260,          9744213492,          3518743761;
  5:      612220032,       9285337152,       60354691488,       217947497040,       472219576920,        613885449996,        443361713886,        137231006679;
  6:    11019960576,     191012649984,     1448512595712,      6276887914752,     16999904769120,      29466501599808,      31922043399792,      19761264961776,       5352009260481;
  7:   198359290368,    3868006162176,    33522720072192,    169475973698304,    550796914519488,    1193393314792224,    1723790343588768,    1600662461903856,     867025500197922,     208728361158759;
  8:  3570467226624,   77360123243520,   754261201624320,   4357953609384960,  16523907435584640,   42962159332520064,   77570565461494560,   96039747714231360,   78032295017812980,   37571105008576620,    8140406085191601;
  9: 64268410079232, 1531730440221696, 16593746435735040, 107859351832277760, 467390524606536960, 1417751257973162112, 3071794392275184576, 4753967511854452320, 5150131471175656680, 3719539395849085380, 1611800404867936998, 317475837322472439;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (18+57*x+39*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1404*x + 1521*x^2.
%t Mathematica
P[0, x_] := 324 + 1404*x + 1521*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + 57*x + 39*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1404*x + 1521*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + 57*x + 39*x^2)*v[i - 1])/(1 + 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

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.