Real-rooted atlas

A013610 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(16 + 24\,x + 9\,x^{2} + 12\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
24042991
320836018927784
4102424962160756816517
54864153601872010800283524352822
622528875521382401123204860010206729420175
71024004730889192969676805896802041203572121873294172
845875224576005677056735436858060802830464816480122472656125529833
92031616123863043317760051093504496419843135283212737088314928041334319683196003234
1089128966094848018579456033177600038320128029785190415676416054587520118098001377810590491493083459
11387973122941255681005649920204374016027371520002529128448163818547273903104022517352043302600454677317714711299009960
121677721601396703232529426022412067799040183936614401970749440015174770688842495385633256396809006940801558893601488034853144185025049949
13721420288654311424027235713024688253829121176610406401434705592321280987136008454515097641071650048144111052803512706912552698640483611311594323636699211246
1430870077443029965209613740539904038129998233672266652057698835274137610042939146247685922816004438620426241916677002246052664217613412153664193444524015624365447829694747561509943
Technical details
Node id
n-4bf16590d3e16b9c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-87f28ed98a2aba0a

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,        3;
  2:      40,       42,        9;
  3:     208,      360,      189,       27;
  4:    1024,     2496,     2160,      756,       81;
  5:    4864,    15360,    18720,    10800,     2835,      243;
  6:   22528,    87552,   138240,   112320,    48600,    10206,      729;
  7:  102400,   473088,   919296,   967680,   589680,   204120,    35721,    2187;
  8:  458752,  2457600,  5677056,  7354368,  5806080,  2830464,   816480,  122472,   6561;
  9: 2031616, 12386304, 33177600, 51093504, 49641984, 31352832, 12737088, 3149280, 413343, 19683;
%F Formula
The row polynomials satisfy (3*n+1+3*x)*P(n,x) = (12*n+16+(9*n+24)*x+9*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[((12*n + 16 + (9*n + 24)*x + 9*x^2)*P[n - 1, x])/(3*n + 1 + 3*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] = ((12*n + 16 + (9*n + 24)*x + 9*x^2)*v[i - 1])/(3*n + 1 + 3*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.