Real-rooted atlas

A038327 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 13\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131316
2978169256
327351152121974096
48114049126263642856165536
52435265456301977304284153712931048576
672918954205335118638038557356683274482680916777216
72187663398624076228495269901457017437710136298962748517268435456
86561227448344962829896776161940870561395016121635586815059644088157307214294967296
9196837676371330570813453549287448069837894163581094720281220330519508220247294671060449937368719476736
1059049255879049896405576580680437240349022736498148821040210902033051950803303709420053181349811901378584918491099511627776
1117714784440071829534852378395305206127593101250507398145418865391941677267859410363408036205552492271896354549330231017179216039403717592186044416
12531441276349326586325469513581220927574168956431180904723251319235164120763285877523270672325849562990726275620818879441583066451777418533223298085122481281474976710656
1315943238981352923351517543710296675840194880654531352006910511811449288162878496135820388255112441416261614159581187295106454327405797812580965966139749086253197767593028751065922534503599627370496
1447829692901667868173031139141665873076168818498748914630936558238951010876285474709768149223281785787089913827515894048197327811177704377608769176133523525956361908113171531193912720754476874626393737638569928972057594037927936
Technical details
Node id
n-7668e44b2e72b4c0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a39bea8eb66cb9fc

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:     3,     13;
  2:     9,     78,      169;
  3:    27,    351,     1521,      2197;
  4:    81,   1404,     9126,     26364,     28561;
  5:   243,   5265,    45630,    197730,    428415,     371293;
  6:   729,  18954,   205335,   1186380,   3855735,    6683274,     4826809;
  7:  2187,  66339,   862407,   6228495,  26990145,   70174377,   101362989,    62748517;
  8:  6561, 227448,  3449628,  29896776, 161940870,  561395016,  1216355868,  1505964408,   815730721;
  9: 19683, 767637, 13305708, 134535492, 874480698, 3789416358, 10947202812, 20330519508, 22024729467, 10604499373;
%F Formula
The row polynomials satisfy P(n,x) = (3+13*x)*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[(3 + 13*x)*P[n - 1, 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] = (3 + 13*x)*v[i - 1]); 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.