Real-rooted atlas

A147716 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(225 - 60\,x + 180\,n x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(3375 + 2700\,x + 720\,x^{2} + 64\,x^{3} + 2700\,n x + 1200\,n x^{2} + 128\,n x^{3} + 240\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1151631
2225480144849
33375108006480102421679
450625216000194400614406400528865
57593754050000486000023040004800003686412490239
6113906257290000010935000069120000216000003317760200704287879089
71708593751275750000229635000018144000007560000001741824002107392010485766509664271
8256289062521870000000459270000004354560000022680000000696729600012644352001258291205308416144948359361
938443359375369056250000885735000000979776000000612360000000235146240000568995840008493465600716636160262144003186652749535
1057665039062561509375000001660753125000020995200000000153090000000007054387200000213373440000042467328000053747712000393216000012687769669309920770321
11864975585937510149046875000030447140625000043302600000000036085500000000019399564800000070413235200000175177728000002956124160000324403200000209348198406039797761493721353018991
12129746337890625166075312500000054804853125000008660520000000000811923750000000049884595200000002112397056000000630639820800000133025587200000194641920000001884133785600108716359680283534950431936724136885409
1319461950683593752698723828125000097154057812500000168880140000000000175916812500000000121593700800000000588453465600000002049579417600000051879979008000009488793600000001224686960640001059984506880055289315328013153337344678089797046532799
1429192926025390625435947695312500000170019601171875000032240754000000000003694253062500000000283718635200000000015446903472000000006148738252800000001815799265280000003985293312000000064296065433600007419891548160005805378109440027622008422406039797760014309080951762730865
Technical details
Node id
n-5f871260d2ebe306 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e4d4f060223d7e66

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,           16;
  2:         225,          480,          144;
  3:        3375,        10800,         6480,         1024;
  4:       50625,       216000,       194400,        61440,         6400;
  5:      759375,      4050000,      4860000,      2304000,       480000,        36864;
  6:    11390625,     72900000,    109350000,     69120000,     21600000,      3317760,      200704;
  7:   170859375,   1275750000,   2296350000,   1814400000,    756000000,    174182400,    21073920,    1048576;
  8:  2562890625,  21870000000,  45927000000,  43545600000,  22680000000,   6967296000,  1264435200,  125829120,   5308416;
  9: 38443359375, 369056250000, 885735000000, 979776000000, 612360000000, 235146240000, 56899584000, 8493465600, 716636160, 26214400;
%F Formula
The row polynomials satisfy (225+(180*n-60)*x+16*n^2*x^2)*P(n,x) = (3375+(2700*n+2700)*x+(240*n^2+1200*n+720)*x^2+(64*n^2+128*n+64)*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 + (2700*n + 2700)*x + (240*n^2 + 1200*n + 720)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(225 + (180*n - 60)*x + 16*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 + (2700*n + 2700)*x + (240*n^2 + 1200*n + 720)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(225 + (180*n - 60)*x + 16*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.