Real-rooted atlas

A038220 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 10\,x + 30\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(125 + 150\,x + 60\,x^{2} + 8\,x^{3} + 150\,n x + 100\,n x^{2} + 16\,n x^{3} + 20\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15813
2258036141
31256005401281393
462540005400256040012985
53125250004500032000100001152116277
6156251500003375003200001500003456031361010821
77812587500023625002800000175000060480010976081928588377
83906255000000157500002240000017500000806400021952003276802073671648241
91953125281250001012500001680000001575000009072000032928000737280093312051200588833245
109765625156250000632812500120000000013125000009072000004116000001228800002332800025600001239044779020029
1148828125859375000386718750082500000001031250000083160000004527600000168960000042768000070400000681472029491238376280257
1224414062546875000002320312500055000000000773437500007128000000045276000000202752000006415200000140800000020444160017694720692224305355744169
13122070312525390625000137109375000357500000000558593750000579150000000420420000000219648000000833976000002288000000044295680005750784004499456016056322410361299717
1461035156251367187500007998046875002275000000000391015625000045045000000003678675000000219648000000097297200000032032000000077517440000134184960001574809600112394240368640018893357029365
Technical details
Node id
n-591fac557f6e52d1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2e8ee919cea99b33

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:       5,        8;
  2:      25,       80,        36;
  3:     125,      600,       540,       128;
  4:     625,     4000,      5400,      2560,       400;
  5:    3125,    25000,     45000,     32000,     10000,     1152;
  6:   15625,   150000,    337500,    320000,    150000,    34560,     3136;
  7:   78125,   875000,   2362500,   2800000,   1750000,   604800,   109760,    8192;
  8:  390625,  5000000,  15750000,  22400000,  17500000,  8064000,  2195200,  327680,  20736;
  9: 1953125, 28125000, 101250000, 168000000, 157500000, 90720000, 32928000, 7372800, 933120, 51200;
%F Formula
The row polynomials satisfy (25+(30*n-10)*x+4*n^2*x^2)*P(n,x) = (125+(150*n+150)*x+(20*n^2+100*n+60)*x^2+(8*n^2+16*n+8)*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[((125 + (150*n + 150)*x + (20*n^2 + 100*n + 60)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(25 + (30*n - 10)*x + 4*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] = ((125 + (150*n + 150)*x + (20*n^2 + 100*n + 60)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(25 + (30*n - 10)*x + 4*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

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.