Real-rooted atlas

A013620 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-7 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(25 + 120\,x + 144\,x^{2} + 60\,n + 144\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1171229
2145408144697
3102552207344172815317
466254920012528011750420736319345
5406253975001476000250560017625602488326431117
62406252925000143100003542400045100800253808642985984126367273
7139062520212500122850000400680000743904000757693440355332096358318082437894469

All 15 rows

Technical details
Node id
n-b9dbb3870c0680fd · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-807e816a03f99168

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:       17,        12;
  2:      145,       408,        144;
  3:     1025,      5220,       7344,        1728;
  4:     6625,     49200,     125280,      117504,        20736;
  5:    40625,    397500,    1476000,     2505600,      1762560,       248832;
  6:   240625,   2925000,   14310000,    35424000,     45100800,     25380864,      2985984;
  7:  1390625,  20212500,  122850000,   400680000,    743904000,    757693440,    355332096,     35831808;
  8:  7890625, 133500000,  970200000,  3931200000,   9616320000,  14282956800,  12123095040,   4873125888,   429981696;
  9: 44140625, 852187500, 7209000000, 34927200000, 106142400000, 207712512000, 257093222400, 187042037760, 65787199488, 5159780352;
%F Formula
The row polynomials satisfy (12*n-7+12*x)*P(n,x) = (60*n+25+(144*n+120)*x+144*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[((60*n + 25 + (144*n + 120)*x + 144*x^2)*P[n - 1, x])/(12*n - 7 + 12*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] = ((60*n + 25 + (144*n + 120)*x + 144*x^2)*v[i - 1])/(12*n - 7 + 12*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.