Real-rooted atlas

A038303 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + x + n\right)\,P_{n}(x) = \left(100 + 20\,x + x^{2} + 10\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111112
2120221143
313003603311694
414000520072044119965
515000070000130001200551234256
616000009000002100002600018006612737867
71700000011200000315000049000045500252077131888098
8180000000136000000448000008400000980000728003360881370256249
91900000000162000000061200000013440000018900000176400010920043209914287177620
102000000000019000000000810000000020400000003360000003780000029400001560005400110149516901511
1121000000000022000000000010450000000029700000000561000000073920000069300000462000021450066001211570623341222
12220000000000025200000000001320000000000418000000000891000000001346400000014784000001188000006930000286000792013216562168424053
1323000000000000286000000000001638000000000057200000000001358500000000231660000000291720000002745600000193050000100100003718009360143175322281041304
142400000000000003220000000000002002000000000007644000000000020020000000000380380000000054054000000058344000000480480000030030000014014000473200109201541863067803598275
Technical details
Node id
n-e587142220425fee · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4477d22738b2090e

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:         11,          1;
  2:        120,         22,         1;
  3:       1300,        360,        33,         1;
  4:      14000,       5200,       720,        44,        1;
  5:     150000,      70000,     13000,      1200,       55,       1;
  6:    1600000,     900000,    210000,     26000,     1800,      66,      1;
  7:   17000000,   11200000,   3150000,    490000,    45500,    2520,     77,    1;
  8:  180000000,  136000000,  44800000,   8400000,   980000,   72800,   3360,   88,  1;
  9: 1900000000, 1620000000, 612000000, 134400000, 18900000, 1764000, 109200, 4320, 99, 1;
%F Formula
The row polynomials satisfy (n+9+x)*P(n,x) = (10*n+100+(n+20)*x+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[((10*n + 100 + (n + 20)*x + x^2)*P[n - 1, x])/(n + 9 + 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] = ((10*n + 100 + (n + 20)*x + x^2)*v[i - 1])/(n + 9 + 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

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.