Real-rooted atlas

A038291 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(10 + 23\,x + 13\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1101323
2100260169529
3100039005070219712167
410000520001014008788028561279841
51000006500001690000219700014280503712936436343
610000007800000253500004394000042841500222775804826809148035889
71000000091000000354900000768950000999635000779715300337876630627485173404825447
81000000001040000000473200000012303200000199927000002079240800013515065200501988136081573072178310985281
91000000000117000000006084000000018454800000035986860000046782918000040545195600022589466120073415764890106044993731801152661463
10100000000001300000000007605000000002636400000000599781000000093565836000001013629890000075298220400003670788244500106044993730013785849184941426511213649
1110000000000014300000000009295000000000362505000000009425130000000017153736600000022299857580000020707010610000013459556896500058324746551500151644341033901792160394037952809757913927
121000000000000156000000000001115400000000004833400000000001413769500000000294064056000000044599715160000004969682546400000403786706895000023329898620600009098660462034002150592472844402329808512248121914624432020321
1310000000000000169000000000000131820000000000062834200000000002042111500000000047785409100000000828280424400000001076764551720000001049845437927000007582217051695000039427528668814000139788510734886003028751065922530302875106592253504036361936467383
14100000000000000182000000000000015379000000000000799708000000000002858956100000000007433285860000000001449490742700000000215352910344000000024496393551630000002123020774474600000137996350340849000065234638342946800021201257461457710042402514922915420393737638569928911592836324538749809
Technical details
Node id
n-e3018af0169eeeab · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-de8557183b81eda9

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:         10,          13;
  2:        100,         260,         169;
  3:       1000,        3900,        5070,         2197;
  4:      10000,       52000,      101400,        87880,        28561;
  5:     100000,      650000,     1690000,      2197000,      1428050,       371293;
  6:    1000000,     7800000,    25350000,     43940000,     42841500,     22277580,      4826809;
  7:   10000000,    91000000,   354900000,    768950000,    999635000,    779715300,    337876630,     62748517;
  8:  100000000,  1040000000,  4732000000,  12303200000,  19992700000,  20792408000,  13515065200,   5019881360,   815730721;
  9: 1000000000, 11700000000, 60840000000, 184548000000, 359868600000, 467829180000, 405451956000, 225894661200, 73415764890, 10604499373;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (10+23*x+13*x^2)*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[Together[((10 + 23*x + 13*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((10 + 23*x + 13*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. [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.