Real-rooted atlas

A027467 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(16\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
212832160
3307215361924800
46553649152122881024128000
5131072013107204915208192051203200000
625165824314572801572864039321604915202457676800000
74697620487046430724404019201468006402752512027525121146881792000000
885899345921503238553611274289152469762048011744051201761607681468006452428840960000000
91546188226563092376453122705829396481352914698244227858432084557168641056964608754974722359296921600000000
10274877906944061847529062406184752906240360777252864013529146982403382286745605637144576060397977603774873601048576020480000000000
1148378511622144120946279055360136064563937280907097092915203968549781504011905649344512248034361344035433480192033218887680184549376046137344450560000000000
1284442493013196823221685578629122902710697328640217703302299648010885165114982403809807790243849524519475609617008070492160212600881152017716740096088583700482013265929830400000000000
1314636698788954112439100963668623366037638250443571250313652087029760283014292989542401132057171958169633018334182113287075357324738561105524581990401228360646656092127048499241875931136872415232212992000000000000
142522015791327477768196551321814302721229482698272145408112702580674946662470439112921841664031697600814828748810565866938276249626414667345690624495275012731699268788196212736068788196212736469010428723219542101196837580963844587520000000000000
Technical details
Node id
n-3ca3059e0c8c21b8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-03d7b8417365c509

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            0;
  1:            4;
  2:          128,           32;
  3:         3072,         1536,          192;
  4:        65536,        49152,        12288,         1024;
  5:      1310720,      1310720,       491520,        81920,        5120;
  6:     25165824,     31457280,     15728640,      3932160,      491520,      24576;
  7:    469762048,    704643072,    440401920,    146800640,    27525120,    2752512,     114688;
  8:   8589934592,  15032385536,  11274289152,   4697620480,  1174405120,  176160768,   14680064,   524288;
  9: 154618822656, 309237645312, 270582939648, 135291469824, 42278584320, 8455716864, 1056964608, 75497472, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (16*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 4*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((16*n + 4*n*x)*v[i - 1])/(n - 1)); 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.