Real-rooted atlas

A038323 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
192029
281360400841
3729486010800800024389
4656158320194400288000160000707281
559049656100291600064800007200000320000020511149
653144170858803936600011664000019440000017280000064000000594823321
74782969744017404960116001837080000408240000054432000004032000000128000000017249876309
843046721765275040595213920026453952000734832000001306368000001451520000009216000000025600000000500246412961
93874204897748409780688747536003571283520001190427840000264539520000039191040000003732480000000207360000000051200000000014507145975869
103486784401774840978007748409780004591650240000178564176000004761711360000088179840000000111974400000000933120000000004608000000000010240000000000420707233300201
11313810596097670925682208523250758000568216717200002525407632000007856823744000001745960832000000277136640000000030792960000000002280960000000000101376000000000020480000000000012200509765705829
122824295364817531454306160920511081864006818600606400003409300303200000121219566336000003142729497600000059861514240000000831409920000000008211456000000000054743040000000000221184000000000004096000000000000353814783205469041
13254186582832973431679485060979089059800800797776270948800044320903941600000177283615766400000525284787456000000116729952768000000019454992128000000002401850880000000000213497856000000000012939264000000000004792320000000000008192000000000000010260628712958602189
1422876792454961711722431932120102804351279084009138164558140800055844338966416000024819706207296000008273235402432000000210113914982400000004085548346880000000060526642176000000000672518246400000000005434490880000000000030191616000000000000103219200000000000001638400000000000000297558232675799463481
Technical details
Node id
n-f8eeff0af924feac · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9eb007f6301418ee

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         9,         20;
  2:        81,        360,         400;
  3:       729,       4860,       10800,         8000;
  4:      6561,      58320,      194400,       288000,        160000;
  5:     59049,     656100,     2916000,      6480000,       7200000,       3200000;
  6:    531441,    7085880,    39366000,    116640000,     194400000,     172800000,      64000000;
  7:   4782969,   74401740,   496011600,   1837080000,    4082400000,    5443200000,    4032000000,    1280000000;
  8:  43046721,  765275040,  5952139200,  26453952000,   73483200000,  130636800000,  145152000000,   92160000000,   25600000000;
  9: 387420489, 7748409780, 68874753600, 357128352000, 1190427840000, 2645395200000, 3919104000000, 3732480000000, 2073600000000, 512000000000;
%F Formula
The row polynomials satisfy P(n,x) = (9+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(9 + 20*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; for(n = 1, N, my(i = n + 1); v[i] = (9 + 20*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. [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.