Real-rooted atlas

A147716 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(225 - 30\,x + 90\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(3375 + 1350\,x + 180\,x^{2} + 8\,x^{3} + 1350\,n x + 300\,n x^{2} + 16\,n x^{3} + 60\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115823
222524036501
333755400162012810523
450625108000486007680400215305
5759375202500012150002880003000011524318527
611390625364500002733750086400001350000103680313685274941
717085937563787500057408750022680000047250000544320032928081921662652547
82562890625109350000001148175000054432000001417500000217728000197568009830402073632078829201
93844335937518452812500022143375000012247200000038272500000734832000088905600066355200279936051200613456316135
10576650390625307546875000041518828125002624400000000956812500000220449600000333396000003317760000209952000768000012390411642539169029
11864975585937550745234375000761178515625005412825000000022553437500000606236400000011002068000001368576000001154736000063360000020444160294912219506159395947
1212974633789062583037656250000013701213281250001082565000000000507452343750000155889360000000330062040000004926873600000519631200000380160000001839974400530841606922244114643550816409
1319461950683593751349361914062500024288514453125000211100175000000001099480078125000037998031500000009194585400000001601233920000002026561680000018532800000001195983360005175705600134983680160563276734775832790287
1429192926025390625217973847656250000425049002929687500403009425000000000230890816406250000886620735000000002413578667500000048037017600000007092965880000007783776000000062789126400003622993920001417328640033718272036864001424511370026765645
Technical details
Node id
n-cf9aeea1b9c52e7c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ddf2baab3dc16fba

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:          15,            8;
  2:         225,          240,           36;
  3:        3375,         5400,         1620,          128;
  4:       50625,       108000,        48600,         7680,         400;
  5:      759375,      2025000,      1215000,       288000,       30000,       1152;
  6:    11390625,     36450000,     27337500,      8640000,     1350000,     103680,      3136;
  7:   170859375,    637875000,    574087500,    226800000,    47250000,    5443200,    329280,     8192;
  8:  2562890625,  10935000000,  11481750000,   5443200000,  1417500000,  217728000,  19756800,   983040,   20736;
  9: 38443359375, 184528125000, 221433750000, 122472000000, 38272500000, 7348320000, 889056000, 66355200, 2799360, 51200;
%F Formula
The row polynomials satisfy (225+(90*n-30)*x+4*n^2*x^2)*P(n,x) = (3375+(1350*n+1350)*x+(60*n^2+300*n+180)*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[((3375 + (1350*n + 1350)*x + (60*n^2 + 300*n + 180)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(225 + (90*n - 30)*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] = ((3375 + (1350*n + 1350)*x + (60*n^2 + 300*n + 180)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(225 + (90*n - 30)*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

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.