Real-rooted atlas

A038322 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(19\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
188
2304128432
386647296153617496
421948827724811673616384629856
5521284087795205544960155648016384021257640
611885275225021632021070848088719360186777601572864688747536
7263456933666557541127006056960393322496012420710402091909121468006421695547384
85720779129616861243750421298413158414946254848062931599360158985093122231369728134217728669462604992
91222816538952411896097331260700477501445111619158016269032587264090621503078419078211174422951231488120795955220334926626632
1025815015822320978253231161601647584389324801618679400038401022323831603204304521396224012082867077120218036699136022951231488010737418240610047798798960
11539533830686488227172139236416043043142171110404832914208686080356109468008448017992899436216326313298047795201518989003980802398403690496022441204121609448928051218118419624329112
12111830648542290245179524774590284810904262683347968013773805494755328011598994100846592068373017857622016287886390979461128658237322690560182278680477696025582972698624021543555956736824633720832533669814389330208
13230184751582880744116303874483981849626933528827869480963780144396893962240358118942863638528024125907729760911361185132309532114944427716923740913664112557085194977280210633141885337602660629160656896203684529045504714682558054415609842070887908584
14470993414777279060825780692177282643328651301697110298357761005518409573793955841058440431130309427208021864320145503027245035027762220367872189621169525138391045988036932372791296140071039353749504023590911891157811227090042363052032190105560442470461572651155456453886177138125341904
Technical details
Node id
n-04dc627a0f0084ad · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-fa8b89dcb83e73ac

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:             8;
  2:           304,           128;
  3:          8664,          7296,          1536;
  4:        219488,        277248,        116736,         16384;
  5:       5212840,       8779520,       5544960,       1556480,        163840;
  6:     118852752,     250216320,     210708480,      88719360,      18677760,      1572864;
  7:    2634569336,    6655754112,    7006056960,    3933224960,    1242071040,    209190912,     14680064;
  8:   57207791296,  168612437504,  212984131584,  149462548480,   62931599360,  15898509312,   2231369728,   134217728;
  9: 1222816538952, 4118960973312, 6070047750144, 5111619158016, 2690325872640, 906215030784, 190782111744, 22951231488, 1207959552;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (19*n+8*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 8*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] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((19*n + 8*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.