Real-rooted atlas

A038327 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + n x\right)\,P_{n}(x) = \left(169 + 26\,x + x^{2} + 13\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113215
2169523224
32197101411743332
428561175763042208549392
53712932856106591067603256729904
648268094455516128524517576012675468710756480
76274851767575326233914593998540384475212946378158120256
881573072110039762724054519568316963299963507381923312483292319097088
91060449937314683152978677683983616218078242339145902159211612918364867210531033943148288
1013785849184921208998746011012364733530119288160506814945056139501641984670210912068445130011495865122816
11179216039403730328868206781749742396545538382275860103535053050133799145481200761562754010403262545929501573127231366374400
12232980851224814301184945688827295981386102933195944824020189335344752981809527843121980061223525124481272392554833400122694187213105288694411264
1330287510659225360575021318450641936553220465815771011467525637911085258475629907262756275373518620466262433952430068681920421115069117621581842197141530735634132992
1439373763856992898480502984583084636037723843731326093855337178726899817517042451273812464684761714747548614117228232827521304541668437433285860314485171959649619992725481522224013651116032
Technical details
Node id
n-f8bab449ff892a5c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f4458aac68149b19

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:          13,           2;
  2:         169,          52,          3;
  3:        2197,        1014,        117,          4;
  4:       28561,       17576,       3042,        208,         5;
  5:      371293,      285610,      65910,       6760,       325,        6;
  6:     4826809,     4455516,    1285245,     175760,     12675,      468,       7;
  7:    62748517,    67575326,   23391459,    3998540,    384475,    21294,     637,     8;
  8:   815730721,  1003976272,  405451956,   83169632,   9996350,   738192,   33124,   832,    9;
  9: 10604499373, 14683152978, 6776839836, 1621807824, 233914590, 21592116, 1291836, 48672, 1053, 10;
%F Formula
The row polynomials satisfy (13+n*x)*P(n,x) = (169+(13*n+26)*x+(n+1)*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[((169 + (13*n + 26)*x + (n + 1)*x^2)*P[n - 1, x])/(13 + n*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] = ((169 + (13*n + 26)*x + (n + 1)*x^2)*v[i - 1])/(13 + n*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.