Real-rooted atlas

A038324 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(21\,n + 10\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 10\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11010
2420200620
31323012600300028830
4370440529200252000400001191640
597240501852200013230000420000050000046176050
62450460605834430005556600002646000006300000060000001717749060
7600362847017153224200204205050001296540000046305000008820000007000000062125257670
814408708328048029027760068612896800054454680000025930800000074088000000117600000008000000002201009128880
934040573424901296783749520021613062492000205838690400001225230300000046675440000001111320000000151200000000900000000076760193369690
1079428004658100340405734249000648391874760000720435416400000514596726000000245046060000000777924000000001587600000000018900000000001000000000002643962216067100
1118347869076021108737080512391000187223153836950002377436874120000019811973951000000113211279720000004492511100000000122245200000000021829500000000023100000000000110000000000090159111567888110
124203330006506652022017442891225320052422483074346000074889261534780000071323106223600000047548737482400000022642255944000000077014476000000000183367800000000002910600000000000277200000000000120000000000003049017227568579720
139562575764802633305464329008458647600143113378792964580002271640933221660000024339009998803500000185440076181360000001030222645452000000042049903896000000001251485235000000000264864600000000000378378000000000003276000000000000130000000000000102396161892511468930
1421626132883476724540133876060707236866200382503030592105332000667862434367168040000795074326627581000000681492279966498000000432693511089840000000206044529090400000000735873318180000000001946754810000000000037081044000000000004815720000000000003822000000000000014000000000000003418456481642305962740
Technical details
Node id
n-60cc7844ab862bfe · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-09bb967aa8554943

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:           420,            200;
  3:         13230,          12600,           3000;
  4:        370440,         529200,         252000,          40000;
  5:       9724050,       18522000,       13230000,        4200000,         500000;
  6:     245046060,      583443000,      555660000,      264600000,       63000000,       6000000;
  7:    6003628470,    17153224200,    20420505000,    12965400000,     4630500000,     882000000,      70000000;
  8:  144087083280,   480290277600,   686128968000,   544546800000,   259308000000,   74088000000,   11760000000,    800000000;
  9: 3404057342490, 12967837495200, 21613062492000, 20583869040000, 12252303000000, 4667544000000, 1111320000000, 151200000000, 9000000000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (21*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[((21*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] = ((21*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.