Real-rooted atlas

A013622 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 - 120\,x + 360\,n x + 225\,n^{2} x^{2}\right)\,P_{n}(x) = \left(512 + 2880\,x + 5400\,x^{2} + 3375\,x^{3} + 2880\,n x + 9000\,n x^{2} + 6750\,n x^{3} + 1800\,n^{2} x^{2} + 3375\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
186068
26496020253049
3512115204860054000114632
44096122880777600172800012656253898201
532768122880010368000345600005062500027337500124152068
626214411796480124416000552960000121500000013122000005581406253774775249
720971521101004801393459200774144000022680000000367416000003125587500010935000000110859571832
8167772161006632960148635648009909043200036288000000078382080000010001880000006998400000002075941406253169300347601
913421772890596966401528823808001189085184000522547200000014108774400000240045120000002519424000000014946778125000384433593750088675273941668
1010737418248053063680015288238080001358954496000069672960000000225740390400000480090240000000671846400000000597871125000000307546875000000697746972656252437742660812249
11858993459270866960384014948499456000149484994560000875888640000000331085905920000084495882240000001478062080000000017537553000000000135320625000000006140173359375000124556484375000066037461179879432
1268719476736618475290624014350559477760015945066086400001051066368000000045406067097600000135193411584000000283787919360000000420901272000000000433026000000000000294728321250000000119574225000000000219271311035156251766799276750916201
135497558138885360119185408013567801688064001658286872985600012145655808000000059027887226880000020085878292480000004918990602240000000875474645760000000011258676000000000000102172484700000000006217859700000000000228042163476562500038145423339843750046767714157447192868
144398046511104461794883665920126632815755264001688437543403520001360313450496000000734569263267840000028120229609472000000787038496358400000001634219338752000000002521943424000000000002860829571600000000002321334288000000000001277036115468750000004272287414062500000065684083557128906251226539614835745346049
Technical details
Node id
n-b7e42fe8311c70bc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2ead7e86fdf601f9

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,         60;
  2:        64,        960,         2025;
  3:       512,      11520,        48600,         54000;
  4:      4096,     122880,       777600,       1728000,       1265625;
  5:     32768,    1228800,     10368000,      34560000,      50625000,       27337500;
  6:    262144,   11796480,    124416000,     552960000,    1215000000,     1312200000,      558140625;
  7:   2097152,  110100480,   1393459200,    7741440000,   22680000000,    36741600000,    31255875000,    10935000000;
  8:  16777216, 1006632960,  14863564800,   99090432000,  362880000000,   783820800000,  1000188000000,   699840000000,   207594140625;
  9: 134217728, 9059696640, 152882380800, 1189085184000, 5225472000000, 14108774400000, 24004512000000, 25194240000000, 14946778125000, 3844335937500;
%F Formula
The row polynomials satisfy (64+(360*n-120)*x+225*n^2*x^2)*P(n,x) = (512+(2880*n+2880)*x+(1800*n^2+9000*n+5400)*x^2+(3375*n^2+6750*n+3375)*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 + (2880*n + 2880)*x + (1800*n^2 + 9000*n + 5400)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*P[n - 1, x])/(64 + (360*n - 120)*x + 225*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 + (2880*n + 2880)*x + (1800*n^2 + 9000*n + 5400)*x^2 + (3375*n^2 + 6750*n + 3375)*x^3)*v[i - 1])/(64 + (360*n - 120)*x + 225*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.