Real-rooted atlas

A038255 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 2\,n x\right)\,P_{n}(x) = \left(49 + 28\,x + 4\,x^{2} + 14\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2495612117
3343588252321215
42401548835288968012393
5168074802041160156802800192124659
61176494033684321802195205880080644481240029
782354332941724235364268912096040019756821952102412223143
85764801263533763953006430118144134456003687936614656573442304119574225
9403536072075328363557705763162405121694145605808499212907776180633614515251201162261467
1028247524916141442803112992540316240512019765032008131898882258860804214784050803203584001126411235194181
111977326743124289109562663338062030438149280217415352001043593689634786456328113459201303948801379840086732824576108090316431
12138412872019491168366422372039720828408939328022828611960012523124275248701038848136306114562738292480386355200364277762064384532481035574967097
1396889010407719746934452185077783144825852134788482308226320400142450538630463311350502420673094041649836923136878958080011049758729392947248455681146889885033776835
14678223072849542578458279215114685623492230319019024642262061793992015511280873088775564043654428942331658248140030778881722757836802707190886430683627522374328321123942424576094049035648173
Technical details
Node id
n-e03304a3f36d13e4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bb158f370ed9f39a

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:        7,         4;
  2:       49,        56,        12;
  3:      343,       588,       252,        32;
  4:     2401,      5488,      3528,       896,        80;
  5:    16807,     48020,     41160,     15680,      2800,      192;
  6:   117649,    403368,    432180,    219520,     58800,     8064,      448;
  7:   823543,   3294172,   4235364,   2689120,    960400,   197568,    21952,    1024;
  8:  5764801,  26353376,  39530064,  30118144,  13445600,  3687936,   614656,   57344,   2304;
  9: 40353607, 207532836, 355770576, 316240512, 169414560, 58084992, 12907776, 1806336, 145152, 5120;
%F Formula
The row polynomials satisfy (7+2*n*x)*P(n,x) = (49+(14*n+28)*x+(4*n+4)*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[((49 + (14*n + 28)*x + (4*n + 4)*x^2)*P[n - 1, x])/(7 + 2*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] = ((49 + (14*n + 28)*x + (4*n + 4)*x^2)*v[i - 1])/(7 + 2*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

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.