Real-rooted atlas

A038324 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(21 + 52\,x + 31\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213152
244113029612704
39261410136054329791140608
41944811148364254280625024449235217311616
5408410130144555889982101313783109696970528629151380204032
685766121759642786280344361555178890206109091415360727302688750368119770609664
718010885411861124825782421242281202782421485299345479335265134567411130463041107275126141111028071702528
83782285936144666995816823077947838686813489361896125725101320701484753577501610958895452988462211917064885289103744153459728531456
97942800465811055257776171962310459164436214624914899724475240882992246701546065369506690410413538244436790261626236161196406076349264396221606712779905883635712
10166798809782012462268144401101635649553066445643874744699172016633430904728610294649347455192523624654671075781030575318313836520169256226380166455552320653740910819628286980801144555105949057024
113502775005422215687839413566541419816718620387551859188325318859155489032198560441301134399987702491202167459045803701082217657246326240590301303272943127281665641293035507075105189334134292565031254084768964048317516865509350965248
12735582751138664113030323020170621210579381309233766265205727310892805620172904514254653900954083839955728968327270332799237554454524889925214842525751128210619541701874489553868614982594308820238561009208631939066402936177894017412787662783788549761390877006486250192896
131544723777391194612964398487088816323262561008856438017181421163555873735934265244770265724501662151393610384892510441657274297917026462372643640491597275334921675964482998269769223471267367653297256206157696521709051837985506454468740007882825333767382150319399742740847532441754629744504259120325604337285010030592
1432439199325215086816704101193877784607464327447169827314209137983825947898033152041541962458123003488672145524605906488674427462100804484507224921946523170064708556406671039032219666915218691950092083216180138786649220725542159561531009193472440281856520772516882709203243160969517621829045869171787586114488425217547569439352207963203211056931425538820521590784
Technical details
Node id
n-9983456df7a955bf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e18808b9d43caba2

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:           21,             31;
  2:          441,           1302,            961;
  3:         9261,          41013,          60543,           29791;
  4:       194481,        1148364,        2542806,         2502444,          923521;
  5:      4084101,       30144555,       88998210,       131378310,        96969705,        28629151;
  6:     85766121,      759642786,     2803443615,      5517889020,      6109091415,      3607273026,       887503681;
  7:   1801088541,    18611248257,    82421242281,    202782421485,    299345479335,    265134567411,    130463041107,     27512614111;
  8:  37822859361,   446669958168,  2307794783868,   6813489361896,  12572510132070,  14847535775016,  10958895452988,   4622119170648,    852891037441;
  9: 794280046581, 10552577761719, 62310459164436, 214624914899724, 475240882992246, 701546065369506, 690410413538244, 436790261626236, 161196406076349, 26439622160671;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (21+52*x+31*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[((21 + 52*x + 31*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] = ((21 + 52*x + 31*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

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.