Real-rooted atlas

A038271 → x ↦ x/(x+1) → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 12\,n x\right)\,P_{n}(x) = \left(49 + 168\,x + 144\,x^{2} + 84\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
172431
249336432817
334335289072691219855
4240132928127008193536103680459553
516807288120148176033868803628800149299210295359
611764924202081555848047416320762048006270566420901888225325009
7823543197650321524731045808499201244678400153628876810241925122866544644845725743
8576480115812025614230823046505519104174254976002867739033628677390336160526499843869835264102795249985
94035360712451970161280774073668307950592219561269760451668897792602225197056505658474496243799621632515978035202156912506207
1028247524996848656801120677314406830795059202561548147200632336456908810538940948480117986977382408532986757120361184624640068109100646444853589991281
1119773267437457346573695880170232065746402444802817702961920081149845303296162299690606592227124931461120219013326766080139056080486400524440074977288916100448256925791004927951
12138412872015694701019848053934299488613633089484802958588110016009737981436395522272195668492288381569884854681645992798620876803893570253619200220264831490457674895243765350411590930582732818987912200409697
1396889010407431848160671266628001932128558406111431168299146131123840011076953883899904295385436903997445787143253629337683706893489995776885787232698368006681366555210547234077335913234432105477468302868481497904875307008387330110836578175
14678223072849325547074967525441286824457124974890810932224293163208501363201206157200691322883618471602073968648102000555081072641367212593669931008173614297608880128016369348060265840641113192973165658112516839594684055552146794677780086784192587769682329607863907907442070033
Technical details
Node id
n-e89238c022dafb97 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f565be5c3ba48d91

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:        7,         24;
  2:       49,        336,         432;
  3:      343,       3528,        9072,        6912;
  4:     2401,      32928,      127008,      193536,       103680;
  5:    16807,     288120,     1481760,     3386880,      3628800,      1492992;
  6:   117649,    2420208,    15558480,    47416320,     76204800,     62705664,     20901888;
  7:   823543,   19765032,   152473104,   580849920,   1244678400,   1536288768,   1024192512,    286654464;
  8:  5764801,  158120256,  1423082304,  6505519104,  17425497600,  28677390336,  28677390336,  16052649984,   3869835264;
  9: 40353607, 1245197016, 12807740736, 68307950592, 219561269760, 451668897792, 602225197056, 505658474496, 243799621632, 51597803520;
%F Formula
The row polynomials satisfy (7+12*n*x)*P(n,x) = (49+(84*n+168)*x+(144*n+144)*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[((49 + (84*n + 168)*x + (144*n + 144)*x^2)*P[n - 1, x])/(7 + 12*n*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] = ((49 + (84*n + 168)*x + (144*n + 144)*x^2)*v[i - 1])/(7 + 12*n*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.