Real-rooted atlas

A013622 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 5\,n x\right)\,P_{n}(x) = \left(16 + 80\,x + 100\,x^{2} + 40\,n x + 100\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
184048
26464012001904
35127680288003200068992
440968192046080010240008000002370816
532768819200614400020480000320000001920000078675968
62621447864320737280003276800007680000009216000004480000002547134464
720971527340032082575360045875200001433600000025804800000250880000001024000000080957571072
8167772166710886408808038400587202560002293760000005505024000008028160000006553600000002304000000002536670560256
913421772860397977609059696640070464307200033030144000009909043200000192675840000002359296000000016588800000000512000000000078582815653888
101073741824536870912009059696640008053063680000440401920000001585446912000003853516800000006291456000000006635520000000004096000000000001126400000000002411887957377024
118589934592472446402560885837004800088583700480000553648128000000232532213760000067821895680000001384120320000000019464192000000000180224000000000009912320000000000245760000000000073456798140465152
1268719476736412316860416085040352460800944892805120000664377753600000031890132172800000108515033088000000265751101440000000467140608000000000576716800000000000475791360000000000235929600000000000532480000000000002222660537282461696
13549755813888357341279027208040178778112009826885173248000767725404160000004145717182464000001612223348736000000460635242496000000097165246464000000001499463680000000000016494100480000000000122683392000000000005537792000000000000114688000000000000066878860345693175808
144398046511104307863255777280750416685957120010005555812761600085985245265920000051591147159552000002257112688230400000073701638799360000000181375126732800000000335879864320000000000461834813440000000000458017996800000000000310116352000000000000128450560000000000000245760000000000000002002650318129367875584
Technical details
Node id
n-7cefd2e3ef7f97d1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2a7fe65055c41fdc

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         8,         40;
  2:        64,        640,        1200;
  3:       512,       7680,       28800,        32000;
  4:      4096,      81920,      460800,      1024000,        800000;
  5:     32768,     819200,     6144000,     20480000,      32000000,      19200000;
  6:    262144,    7864320,    73728000,    327680000,     768000000,     921600000,      448000000;
  7:   2097152,   73400320,   825753600,   4587520000,   14336000000,   25804800000,    25088000000,    10240000000;
  8:  16777216,  671088640,  8808038400,  58720256000,  229376000000,  550502400000,   802816000000,   655360000000,   230400000000;
  9: 134217728, 6039797760, 90596966400, 704643072000, 3303014400000, 9909043200000, 19267584000000, 23592960000000, 16588800000000, 5120000000000;
%F Formula
The row polynomials satisfy (2+5*n*x)*P(n,x) = (16+(40*n+80)*x+(100*n+100)*x^2)*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[((16 + (40*n + 80)*x + (100*n + 100)*x^2)*P[n - 1, x])/(2 + 5*n*x)]];
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] = ((16 + (40*n + 80)*x + (100*n + 100)*x^2)*v[i - 1])/(2 + 5*n*x)); 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.