Real-rooted atlas

A352880 → reverse → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(3\,n + 10\,n^{2} + 9\,n^{3} + 2\,n^{4}\right)\,P_{n}(x) = \left(24\,x^{2} + 28\,n x^{2} - 16\,n^{2} x^{2} - 28\,n^{3} x^{2} - 8\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(12\,n + 38\,n^{2} + 32\,n^{2} x + 32\,n^{3} + 32\,n^{3} x + 8\,n^{4} + 8\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 8\)
\(P_{1}(x) = 30 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
088
1302454
211219264368
342011208401602540
4158457606912307238417712
5600627720462003360010080896124502
622880128128274560281600140800307202048880736
7875165765761513512201801614414405322248870446086262596
8335920254592079206401304576012230400652288018636802457601024044721200
91293292110853603990729678414336914833926457651226906880615014465894422528320498684
10499228847752320195350400446515200625121280552738816307077120103219200193536001720320491522303889696
11193154002039706249348653602438779200399072960042567782402979744768134424576037340160058572800439296010649616604902808
1274884320865329920439321344012886759424241626739203025586995225671647232146695127045501067264128679936017157324811010048229376119950570208
13290845350365061060020339116200662904528001402298040002019309177602019309177601404736819206704425728021283891200425677824048881664027156480491520868237737750
1411314454401532926080092997515520333508331520786126781440128109549568014781871104001216336822272709529812992287925141504785250385921359740928013597409286606028810485766295717015232
Technical details
Node id
n-255ef16c4f62b1af · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-3ac782335c346aee

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:       8;
  1:      30,       24;
  2:     112,      192,       64;
  3:     420,     1120,      840,      160;
  4:    1584,     5760,     6912,     3072,      384;
  5:    6006,    27720,    46200,    33600,    10080,      896;
  6:   22880,   128128,   274560,   281600,   140800,    30720,     2048;
  7:   87516,   576576,  1513512,  2018016,  1441440,   532224,    88704,    4608;
  8:  335920,  2545920,  7920640, 13045760, 12230400,  6522880,  1863680,  245760,  10240;
  9: 1293292, 11085360, 39907296, 78414336, 91483392, 64576512, 26906880, 6150144, 658944, 22528;
%F Formula
The row polynomials satisfy (2*n^4+9*n^3+10*n^2+3*n)*P(n,x) = -(8*n^4+28*n^3+16*n^2-28*n-24)*x^2*P(n-2,x) + (8*n^4+32*n^3+38*n^2+12*n+(8*n^4+32*n^3+32*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 8, P(1,x) = 30 + 24*x.
%t Mathematica
P[0, x_] := 8;
P[1, x_] := 30 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^4 + 28*n^3 + 16*n^2 - 28*n - 24)*x^2*P[n - 2, x] + (8*n^4 + 32*n^3 + 38*n^2 + 12*n + (8*n^4 + 32*n^3 + 32*n^2)*x)*P[n - 1, x])/(2*n^4 + 9*n^3 + 10*n^2 + 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8; v[2] = 30 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^4 + 28*n^3 + 16*n^2 - 28*n - 24)*x^2*v[i - 2] + (8*n^4 + 32*n^3 + 38*n^2 + 12*n + (8*n^4 + 32*n^3 + 32*n^2)*x)*v[i - 1])/(2*n^4 + 9*n^3 + 10*n^2 + 3*n)); 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.