Real-rooted atlas

A038277 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(162 + 396\,x + 242\,x^{2} + 198\,n + 242\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1402262
2111617604843360
327216736565808010648169600
46181922395008324086417036802342568192000
51343692868001120131725440118831680468512005153632384000000
62834352001773674496448807392057959193603921445440123687168011337990417612800000
7585010252843649020800136572936192230387794560223142895360120780519552317463731202494357888794624000000
811877068620810296180449283841113830400801227892326410137062960640785462991667235428952401927981945241605487587353635389440000000
92380311484416235165958691841019321864478722535135128064003966078067015684014276932413442592027872501761002133225082881975531447296012072692177921559756800000000
104720951110758452366852657152025868255456102407475027006177280139432432043520001745074349486899214719015418849280814637331357696027558663689779204829076871168002655992279142468157440000000000
11928321478922240114247016880353286336389171515392020867059401255936045223913387372544067485297109063680070384665429304934450885739019450368024642779273570304074102184588072960116863660282265605843183014113282956984320000000000
12181236916423434242450768704354713601508060622820663296557602247093354496013772259204828917760238782262685327032322969353072798801920026545073819052146688167922938764186214407228548586913955840195629767312512614428047278467743744012855002631049216127506841600000000000
13351676739953557504518337580971021926435045992472272404480143768446042236567552398685606671748464640787773226516214095872113819545213339218739212131928268863676416009489863890311142440965336217831839695257602067364895857391370245086373950125327974466846013681455923202828100578830827525469372416000000000000
146788286546208800768108316435905695711232798239874695373766656359805522715330019328011070170345252215701504245590333709797054218244043902562783232358809650080599893869256245248467079238351251542016003247642309128702079795216435550922066261393408578862170840069583667213055026471988341800961583736324145263411206221821273427820544233538846720000000000000
Technical details
Node id
n-6e71af4ae1c1a2db · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-17467381a9b6e863

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:            40,             22;
  2:          1116,           1760,             484;
  3:         27216,          73656,           58080,           10648;
  4:        618192,        2395008,         3240864,         1703680,          234256;
  5:      13436928,       68001120,       131725440,       118831680,        46851200,         5153632;
  6:     283435200,     1773674496,      4488073920,      5795919360,      3921445440,      1236871680,       113379904;
  7:    5850102528,    43649020800,    136572936192,    230387794560,    223142895360,    120780519552,     31746373120,      2494357888;
  8:  118770686208,  1029618044928,   3841113830400,   8012278923264,  10137062960640,   7854629916672,   3542895240192,    798194524160,    54875873536;
  9: 2380311484416, 23516595869184, 101932186447872, 253513512806400, 396607806701568, 401427693241344, 259202787250176, 100213322508288, 19755314472960, 1207269217792;
%F Formula
The row polynomials satisfy (11*n-2+11*x)*P(n,x) = (198*n+162+(242*n+396)*x+242*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[((198*n + 162 + (242*n + 396)*x + 242*x^2)*P[n - 1, x])/(11*n - 2 + 11*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] = ((198*n + 162 + (242*n + 396)*x + 242*x^2)*v[i - 1])/(11*n - 2 + 11*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.