Real-rooted atlas

A038208 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 24\,x + 48\,x^{2} + 32\,x^{3} + 24\,n x + 80\,n x^{2} + 64\,n x^{3} + 16\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
143236
216256576848
36415366912819216704
4256819255296131072102400297216
51024409603686401310720204800011796484948992
6409619660822118401048576024576000283115521284505678630912
71638491750412386304734003202293760003963617283596615681342177281206337536
8655364194304660602884697620481835008000422785843257545850884294967296135895449618011455488
9262144188743683397386242818572288132120576003805072588869055021056773094113284892236185613421772800263148797952
101048576838860801698693120161061273608808038400030440580710469055021056010307921510409784472371205368709120001299227607043776959217664
114194304369098752830472192088583700480553648128000223230925209660768418529281133871366144014350559477760118111600640005716601470976123695058124853414046203904
1216777216161061273639862665216472446402560332188876800015307263442944486147348234241088516511498241722067137331201889785610240001371984353034245937362789990411613591568384745980404170752
1367108864697932185618844169011224567212933121919313510400099497212379136361138030116864943380976631808179094982282444824567212933120002378106211926016154371432539750460390676155596810775213952204810307012118183936
1426843545630064771072879394553856125069447659521074815565824006190937659146242527966210818048754704781305446416715531679694848275152784850944003329348696696422428816000740753408169093893235671046034119813234688989560464998400141088373493202944
Technical details
Node id
n-9b39ea493a280776 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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:      4,       32;
  2:     16,      256,       576;
  3:     64,     1536,      6912,       8192;
  4:    256,     8192,     55296,     131072,      102400;
  5:   1024,    40960,    368640,    1310720,     2048000,     1179648;
  6:   4096,   196608,   2211840,   10485760,    24576000,    28311552,    12845056;
  7:  16384,   917504,  12386304,   73400320,   229376000,   396361728,   359661568,   134217728;
  8:  65536,  4194304,  66060288,  469762048,  1835008000,  4227858432,  5754585088,  4294967296,  1358954496;
  9: 262144, 18874368, 339738624, 2818572288, 13212057600, 38050725888, 69055021056, 77309411328, 48922361856, 13421772800;
%F Formula
The row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = (4+(24*n+24)*x+(16*n^2+80*n+48)*x^2+(32*n^2+64*n+32)*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[((4 + (24*n + 24)*x + (16*n^2 + 80*n + 48)*x^2 + (32*n^2 + 64*n + 32)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*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] = ((4 + (24*n + 24)*x + (16*n^2 + 80*n + 48)*x^2 + (32*n^2 + 64*n + 32)*x^3)*v[i - 1])/(1 + (6*n - 2)*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.