Real-rooted atlas

A013622 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(8\,n + 15\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 15\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11515
2240450690
32880108001012523805
430720172800324000202500730020
5307200230400064800008100000379687520988075
629491202764800010368000019440000018225000068343750579270870
7275251203096576001451520000362880000051030000003827250000119601562515543768345
8251658240330301440018579456000580608000001088640000001224720000007654500000020503125000408579053640
9226492416033973862400222953472000836075520000195955200000029393280000002755620000000147622500000034599023437510571983012935
102013265920033973862400025480396800001114767360000031352832000000587865600000007348320000000059049000000000276792187500005766503906250270172899219450
111771674009603321888768000280284364800001401421824000004598415360000001034643456000000161663040000000017321040000000001217885625000000507452343750000951473144531256835374350252085
1215461882265603189013217280029896998912000016817061888000006306398208000000165542952960000003103930368000000041570496000000000389723400000000002435771250000000091341421875000001556956054687500171505756424506860
1313400297963520301506704179200310928788684800019433049292800000819831767040000002459495301120000005380145971200000008646663168000000001013280840000000000844400700000000000474975393750000000161923429687500000253005358886718754273351764243962595
141154487209164802814062572339200316582039388160002176501520793600001020235087872000000344329342156800000086082335539200000001614043791360000000022697490816000000000236432196000000000001773241470000000000090677120625000000002833660019531250000408700964355468750105847636006658150430
Technical details
Node id
n-82784edb4f691ed1 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dd0a803b420669cd

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:         15;
  2:        240,         450;
  3:       2880,       10800,        10125;
  4:      30720,      172800,       324000,       202500;
  5:     307200,     2304000,      6480000,      8100000,       3796875;
  6:    2949120,    27648000,    103680000,    194400000,     182250000,      68343750;
  7:   27525120,   309657600,   1451520000,   3628800000,    5103000000,    3827250000,    1196015625;
  8:  251658240,  3303014400,  18579456000,  58060800000,  108864000000,  122472000000,   76545000000,   20503125000;
  9: 2264924160, 33973862400, 222953472000, 836075520000, 1959552000000, 2939328000000, 2755620000000, 1476225000000, 345990234375;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (8*n+15*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 15.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 15;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 15*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] = 15; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 15*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.