Real-rooted atlas

A013622 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 - 40\,x + 120\,n x + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(512 + 960\,x + 600\,x^{2} + 125\,x^{3} + 960\,n x + 1000\,n x^{2} + 250\,n x^{3} + 200\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
182028
264320225609
351238405400200011752
4409640960864006400015625211081
532768409600115200012800006250001125003611868
62621443932160138240002048000015000000540000076562559663929
7209715236700160154828800286720000280000000151200000428750005000000959421112
816777216335544320165150720036700160004480000000322560000013720000003200000003164062515103085361
913421772830198988801698693120044040192000645120000005806080000032928000000115200000002278125000195312500233675477308
10107374182426843545600169869312000503316480000860160000000928972800000658560000000307200000000911250000001562500000011816406253563927520049
1185899345922362232012801660944384000553648128000010813440000000136249344000001159065600000067584000000002673000000000687500000000103984375000703125000053701184824872
1268719476736206158430208015945066086400590558003200001297612800000001868562432000001854504960000001297612800000006415200000000022000000000000499125000000067500000000041259765625800819979150841
13549755813888178670639513601507533520896006141803233280001499463680000000242913116160000027552645120000002249195520000000133436160000000057200000000000017303000000000035100000000000429101562500023925781250011835427242220348
1443980465111041539316278886401407031286169600625347238297600016793993216000000302291877888000003857370316800000035987128320000000249080832000000001281280000000000048448400000000001310400000000000240296875000000267968750000001373291015625173547436077360969
Technical details
Node id
n-15ae131e7f255640 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-472634f5a19ab425

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:         8,         20;
  2:        64,        320,         225;
  3:       512,       3840,        5400,        2000;
  4:      4096,      40960,       86400,       64000,       15625;
  5:     32768,     409600,     1152000,     1280000,      625000,      112500;
  6:    262144,    3932160,    13824000,    20480000,    15000000,     5400000,      765625;
  7:   2097152,   36700160,   154828800,   286720000,   280000000,   151200000,    42875000,     5000000;
  8:  16777216,  335544320,  1651507200,  3670016000,  4480000000,  3225600000,  1372000000,   320000000,   31640625;
  9: 134217728, 3019898880, 16986931200, 44040192000, 64512000000, 58060800000, 32928000000, 11520000000, 2278125000, 195312500;
%F Formula
The row polynomials satisfy (64+(120*n-40)*x+25*n^2*x^2)*P(n,x) = (512+(960*n+960)*x+(200*n^2+1000*n+600)*x^2+(125*n^2+250*n+125)*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[((512 + (960*n + 960)*x + (200*n^2 + 1000*n + 600)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(64 + (120*n - 40)*x + 25*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] = ((512 + (960*n + 960)*x + (200*n^2 + 1000*n + 600)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(64 + (120*n - 40)*x + 25*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.