Real-rooted atlas

A013610 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 36\,x + 108\,n x + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(64 + 432\,x + 972\,x^{2} + 729\,x^{3} + 432\,n x + 1620\,n x^{2} + 1458\,n x^{3} + 324\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
143640
2162887291033
364172887481166422204
4256921669984186624164025430105
51024460804665601866240328050021257647786168
64096221184279936014929920393660005101833626040609134379505
7163841032192156764161045094403674160007142567047291370523061100162238154204
86553647185928360755266886041629393280007618738176116661928329795520512348678440136263816017
9262144212336644299816964013162496211631616006856864358413999431398417631936921612552423843638742048900574776415720
1010485769437184021499084802293235712014108774400054854914867213999431398402350924922880251048476872015496819560004219009125218947750278649
11419430441523609610510663680126127964160886837248000402269375692812319499630592258601741516803682044327456034093003032000185636401509244518872583696137222221886620
121677721618119393285045118566467268247552053210234880002758418576179298555997044736248257671856128441845319294720545488048512000445527363622176216905884017408477305916652892077941047639977
1367108864785173708823849651404834979488727043074369126400017929720745164873213026376089621515664894197764595191320665088709134463065600077224743027843845639552984452608248199076659502849820570235248431126241723634616
14268435456338228674561112983732224178077397155841721646710784001115627068588032512491184632627217212531915358208428884523262074887942305986334720010811464023898137610527165570978201669495741464660784278995193317391045147278302366225461884537553185825
Technical details
Node id
n-a221df859525f5be · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6acd8c5ba29bd39a

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:      4,       36;
  2:     16,      288,       729;
  3:     64,     1728,      8748,      11664;
  4:    256,     9216,     69984,     186624,      164025;
  5:   1024,    46080,    466560,    1866240,     3280500,     2125764;
  6:   4096,   221184,   2799360,   14929920,    39366000,    51018336,     26040609;
  7:  16384,  1032192,  15676416,  104509440,   367416000,   714256704,    729137052,    306110016;
  8:  65536,  4718592,  83607552,  668860416,  2939328000,  7618738176,  11666192832,   9795520512,   3486784401;
  9: 262144, 21233664, 429981696, 4013162496, 21163161600, 68568643584, 139994313984, 176319369216, 125524238436, 38742048900;
%F Formula
The row polynomials satisfy (16+(108*n-36)*x+81*n^2*x^2)*P(n,x) = (64+(432*n+432)*x+(324*n^2+1620*n+972)*x^2+(729*n^2+1458*n+729)*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[((64 + (432*n + 432)*x + (324*n^2 + 1620*n + 972)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*P[n - 1, x])/(16 + (108*n - 36)*x + 81*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] = ((64 + (432*n + 432)*x + (324*n^2 + 1620*n + 972)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*v[i - 1])/(16 + (108*n - 36)*x + 81*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

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.