Real-rooted atlas

A038303 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 33\,x + 99\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 1089\,x + 297\,x^{2} + 27\,x^{3} + 1089\,n x + 495\,n x^{2} + 54\,n x^{3} + 99\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111223
212126481466
31331435626734328792
4146416388858806190082025158368
5161051878460107811052272011137587482760464
6177156111595672177888151149984036753755773683572146944352
71948717114881112427394775122137192094334625222286682750517139968783071744
821435888118707684164017900348389614579220753617506520409281210227481231718453144112860447488
9235794769123150759148568245906366428640556841092162650161380129683993750684609700608526126591968300208507910912
1025937424601282953722920781338121245101021494464075335631525035503628529610982814381022355688960289369624521651300071449293344138000896
11285311670611342374004733210504656963405152795010376801302230202075071598984201362657841080202676259591040116712415215130990365008645364092550916853140212328448
123138428376721410848805679841386614719169462240993485526402148679833423751350152273511365847250376444417853253203456385150970209557635760600057059402994336721017689813529837681480810496
1334522712143931489594826768476180259913492029832046206843027523414013513106625241339718890155611945097197593564255025346823681101531774799172060478441450027198315427142407555275841284333464731248730813112492019372032
14379749833583241579981564018040823133355564810491448646895802385285257580810184202541295907454537736229943121053676039361055763012096282726488865120363462735996660010471351439448912358783749312988936767819481230454321076168025203984696726315008
Technical details
Node id
n-96ffd727c06b8748 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-83385ff7049ec979

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:         11,          12;
  2:        121,         264,          81;
  3:       1331,        4356,        2673,         432;
  4:      14641,       63888,       58806,       19008,        2025;
  5:     161051,      878460,     1078110,      522720,      111375,        8748;
  6:    1771561,    11595672,    17788815,    11499840,     3675375,      577368,      35721;
  7:   19487171,   148811124,   273947751,   221371920,    94334625,    22228668,    2750517,    139968;
  8:  214358881,  1870768416,  4017900348,  3896145792,  2075361750,   652040928,  121022748,  12317184,   531441;
  9: 2357947691, 23150759148, 56824590636, 64286405568, 41092162650, 16138012968, 3993750684, 609700608, 52612659, 1968300;
%F Formula
The row polynomials satisfy (121+(99*n-33)*x+9*n^2*x^2)*P(n,x) = (1331+(1089*n+1089)*x+(99*n^2+495*n+297)*x^2+(27*n^2+54*n+27)*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[((1331 + (1089*n + 1089)*x + (99*n^2 + 495*n + 297)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(121 + (99*n - 33)*x + 9*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] = ((1331 + (1089*n + 1089)*x + (99*n^2 + 495*n + 297)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(121 + (99*n - 33)*x + 9*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.