Real-rooted atlas

A038322 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(19\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 16\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11616
26085121120
317328291841228858800
443897611089929338882621442744000
5104256803511808044359680249036805242880120050000
62377055041000865280168566784014195097605976883201006632965042100000
7526913867226623016448560484556806293159936039746273280133882183681879048192205885750000
81144155825926744497500161703873052672239140077568020138111795201017504595968285615325184343597383688235430000000
924456330779041647584389324848560382001152817859065282568609042792448057997761970176244201103032325875515260928618475290624324270056250000
1051630031644640391301292464640131806751145984025898870400614403271436261130240275489369358336015466069858713605581739497881601175103052185601099511627776012610502187500000
111079067661372976908688556945664034434513736888320773266273389772801139550297627033601151545563917844488081021501177856038886118501908480122798268953395202297979302051840193514046488576485504334218750000
12223661297084580482071809909836113928723410146678374402203808879160852480371167811227090944043758731428878090243684945804537102336221650875460878336093326684404580352026196964043390976044121202599395328337769972052787218537438215625000000
13460369503165761488465215497935927398421546823062295584768604823103503033958401145980617163643289601544058094704698327041516969356201107128321094955324776738979845762922761982836736021568833729058570240544896852102532300883429183097038438458546795155816448702877865675781250000
149419868295545581216103122768709130573312521041357688238686208160882945531807032934433870093796169901670405133993164893121937408576448355356420708761648543019398435428106243065874909374869143552143432744298239492096048314187553091197337611096081351906112307215573447511447175168100880631653099110426493088783164062500000
Technical details
Node id
n-c714928c5eff691f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4b6c87ac223a4827

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:            16;
  2:           608,            512;
  3:         17328,          29184,          12288;
  4:        438976,        1108992,         933888,         262144;
  5:      10425680,       35118080,       44359680,       24903680,        5242880;
  6:     237705504,     1000865280,     1685667840,     1419509760,      597688320,      100663296;
  7:    5269138672,    26623016448,    56048455680,    62931599360,    39746273280,    13388218368,     1879048192;
  8:  114415582592,   674449750016,  1703873052672,  2391400775680,  2013811179520,  1017504595968,   285615325184,   34359738368;
  9: 2445633077904, 16475843893248, 48560382001152, 81785906528256, 86090427924480, 57997761970176, 24420110303232, 5875515260928, 618475290624;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (19*n+16*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 16.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 16;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 16*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] = 16; for(n = 2, N, my(i = n + 1); v[i] = ((19*n + 16*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.