Real-rooted atlas

A038324 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + 10\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 10\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11010
2220200420
336306600300013230
45324014520013200040000370440
57320502662000363000022000005000009724050
69663060439230007986000072600000330000006000000245046060
7124009270676414200153730500018634000001270500000462000000700000006003628470
815589736809920741600270565680004099480000037268000000203280000006160000000800000000144087083280
9192922992901403076312004464333720008116970400009223830000006708240000003049200000007920000000090000000003404057342490
102357947691001929229929000701538156000014881112400000202924260000001844766000000011180400000000435600000000099000000000010000000000079428004658100
11285311670611025937424601000106107646095000257230657200000409230591000000446433372000000338207100000000175692000000000598950000000001210000000000011000000000001834786907602110
123423740047332034237400473320015562454760600004244305843800000771691971600000098215341840000008928667440000000579783600000000026353800000000007986000000000001452000000000001200000000000042033300065066520
13407995688973730445086206153160022254310307658000674373039626000001379399399235000002006399126160000002127999073200000001658181096000000009421483500000000038066600000000000103818000000000001716000000000000130000000000000956257576480263330
14483317970015034057119396456322200311560344307212000103853448102404000023603056386910000003862318317858000000468159796104000000042559981464000000002901816918000000000146556410000000000053293240000000000013213200000000000020020000000000000140000000000000021626132883476724540
Technical details
Node id
n-4d5b6d358f445bab · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c4264000c5d1dd22

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:          10;
  2:         220,          200;
  3:        3630,         6600,         3000;
  4:       53240,       145200,       132000,        40000;
  5:      732050,      2662000,      3630000,      2200000,       500000;
  6:     9663060,     43923000,     79860000,     72600000,     33000000,      6000000;
  7:   124009270,    676414200,   1537305000,   1863400000,   1270500000,    462000000,     70000000;
  8:  1558973680,   9920741600,  27056568000,  40994800000,  37268000000,  20328000000,   6160000000,   800000000;
  9: 19292299290, 140307631200, 446433372000, 811697040000, 922383000000, 670824000000, 304920000000, 79200000000, 9000000000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+10*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 10.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 10;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 10*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] = 10; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 10*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.