Real-rooted atlas

A038221 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 18\,x + 18\,x^{2} + 6\,x^{3} + 18\,n x + 30\,n x^{2} + 12\,n x^{3} + 6\,n^{2} x^{2} + 6\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
162430
236288324648
321625925832345612096
4129620736699848294432400207360
5777615552069984012441609720002799363359232
64665611197446298560149299201749600010077696228614452254720
72799367838208529079041567641602449440002116316169601804817915904788299776
8167961653747712423263232150493593629393280003386105856230443315285996339213604889611609505792
9100776963627970563265173504135444234243174474240045712429056414797967362321901158473466403841007769600167692861440
10604661762418647040244888012801160950579203174474240005485491486726221969510404643802316802203992115206046617600073164072962383818522624
11362797056159630704641795845427209577842278402993075712000603404063539282129997537287662273822720484878265344019953838080004828828815365224277606433435376680960
122176782336104485552128129300870758476622738227202693768140800062064417964032985559970447361103367430471688727808776192047889211392000173837837352963761479876608367876214784463637223309312
13130606940166791560888329168607199232597657358172162334599055360006051280751493121098195395641344143437765961318413615381690859529338396221440004519783771176961466977151877122869434475315225598960271366366095720054784
147836416409643883931893766418025039462445639289169510419610632065024005647862034726912115310516542341121721253191535820819061534367203328156885056520192009491545919471616410753602525593612051624796323842150312662794241763193692160086664496357048320
Technical details
Node id
n-82ea76f4b917d0f7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:        6,        24;
  2:       36,       288,        324;
  3:      216,      2592,       5832,        3456;
  4:     1296,     20736,      69984,       82944,       32400;
  5:     7776,    155520,     699840,     1244160,      972000,      279936;
  6:    46656,   1119744,    6298560,    14929920,    17496000,    10077696,     2286144;
  7:   279936,   7838208,   52907904,   156764160,   244944000,   211631616,    96018048,    17915904;
  8:  1679616,  53747712,  423263232,  1504935936,  2939328000,  3386105856,  2304433152,   859963392,  136048896;
  9: 10077696, 362797056, 3265173504, 13544423424, 31744742400, 45712429056, 41479796736, 23219011584, 7346640384, 1007769600;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (6+(18*n+18)*x+(6*n^2+30*n+18)*x^2+(6*n^2+12*n+6)*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[((6 + (18*n + 18)*x + (6*n^2 + 30*n + 18)*x^2 + (6*n^2 + 12*n + 6)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + 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] = ((6 + (18*n + 18)*x + (6*n^2 + 30*n + 18)*x^2 + (6*n^2 + 12*n + 6)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + 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.