Real-rooted atlas

A038335 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(21\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 9\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
199
2378162540
31190710206218724300
433339642865218370826244972000
58751645150028209644670275562029524536450000
62205414544725888304050761401736040603720087031886461312200000
754032656231389411160214886548145850659894027342639454687309623348078345927000000
81296783749523890351248565001880176723572771554801531187809203937340080856247715443443737681574640000000
930636516082411050394837111215755922556668135050764771447234862398470248052425090453154091090865086642152348678440153144100000000
1071485204192290275728644741690472677676700040472677676700040303864220735740130227523172460372078637635606834097425960732224724210348678440101771470000000000
11165130821684189970770352150367101364856791471365515598363331101320116987724983259906016511570567652214875413234559052622550179892084571955646255805447196631034519165569958458510000000000
123782997005855986817834128741892509238215990161198234049134844492969158042115580993973564025269348596384138410829720827021773633152206613331960710404427428542010148634677550608698829723614833891544377721913187600000000000
13860631818832236997442610649685150455610432965314007117882149042361628673112601437194201419347871598550459525898139764927522976294906988181011048108792501648485102171997991592352575565710460118739025727342029252391615117563304425576827762178597000000000000
141946351959512905208610843960917286186162227884470930164478702843818454318829895104446948343913032030469036217293875767566361820695596500438609349688695413573308325784285092400771348190026787914304079718810116364245212795179613601015674222813297150111958734383202750943694542008846980000000000000
Technical details
Node id
n-cb48487a82c52afe · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5184fe74b2189306

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:             0;
  1:             9;
  2:           378,            162;
  3:         11907,          10206,           2187;
  4:        333396,         428652,         183708,          26244;
  5:       8751645,       15002820,        9644670,        2755620,        295245;
  6:     220541454,      472588830,      405076140,      173604060,      37200870,       3188646;
  7:    5403265623,    13894111602,    14886548145,     8506598940,    2734263945,     468730962,     33480783;
  8:  129678374952,   389035124856,   500188017672,   357277155480,  153118780920,   39373400808,   5624771544,   344373768;
  9: 3063651608241, 10503948371112, 15755922556668, 13505076477144, 7234862398470, 2480524250904, 531540910908, 65086642152, 3486784401;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (21*n+9*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 9.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 9;
P[n_, x_] := P[n, x] = Expand[Together[((21*n + 9*n*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] = 0; v[2] = 9; for(n = 2, N, my(i = n + 1); v[i] = ((21*n + 9*n*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.