Real-rooted atlas

A038291 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(100 - 30\,x + 90\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1000 + 900\,x + 270\,x^{2} + 27\,x^{3} + 900\,n x + 450\,n x^{2} + 54\,n x^{3} + 90\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1101222
210024081421
31000360024304327462
4100004800048600172802025125905
510000060000081000043200010125087482051998
61000000720000012150000864000030375005248803572132588101
7100000008400000017010000015120000070875000183708002500470139968507186238
8100000000960000000226800000024192000001417500000489888000100018800111974405314417766335681
9100000000010800000000291600000003628800000025515000000110224800003000564000503884800478296901968300117339726790
10100000000001200000000003645000000005184000000004252500000002204496000007501410000016796160000239148450019683000071449291753005319429
1110000000000013200000000004455000000000712800000000066825000000004041576000000165031020000046189440000087687765000108256500007859421902550916825938605466358
121000000000000144000000000005346000000000095040000000000100237500000000692841600000003300620400000011085465600000263063295000043302600000047156531400306110016089813529380627295995089
131000000000000015600000000000063180000000000012355200000000001447875000000000112586760000000061297236000000024018508800000068396456700000140733450000002043449694000198971510400116757587703124873085544944259150478
1410000000000000016800000000000007371000000000000157248000000000002027025000000000017513496000000000107270163000000004803701760000000159591732300000039405366000000071520739290000928533715200081730311390043748223120107616802580261903246947045
Technical details
Node id
n-5ce205baf785df9d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-09fd8ade2935dd5d

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         10,          12;
  2:        100,         240,          81;
  3:       1000,        3600,        2430,         432;
  4:      10000,       48000,       48600,       17280,        2025;
  5:     100000,      600000,      810000,      432000,      101250,        8748;
  6:    1000000,     7200000,    12150000,     8640000,     3037500,      524880,      35721;
  7:   10000000,    84000000,   170100000,   151200000,    70875000,    18370800,    2500470,    139968;
  8:  100000000,   960000000,  2268000000,  2419200000,  1417500000,   489888000,  100018800,  11197440,   531441;
  9: 1000000000, 10800000000, 29160000000, 36288000000, 25515000000, 11022480000, 3000564000, 503884800, 47829690, 1968300;
%F Formula
The row polynomials satisfy (100+(90*n-30)*x+9*n^2*x^2)*P(n,x) = (1000+(900*n+900)*x+(90*n^2+450*n+270)*x^2+(27*n^2+54*n+27)*x^3)*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[((1000 + (900*n + 900)*x + (90*n^2 + 450*n + 270)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(100 + (90*n - 30)*x + 9*n^2*x^2)]];
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] = ((1000 + (900*n + 900)*x + (90*n^2 + 450*n + 270)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(100 + (90*n - 30)*x + 9*n^2*x^2)); 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.