Real-rooted atlas

A038271 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(144 - 60\,x + 180\,n x + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1728 + 2160\,x + 900\,x^{2} + 125\,x^{3} + 2160\,n x + 1500\,n x^{2} + 250\,n x^{3} + 300\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1122032
2144480225849
3172886408100200020468
4207361382401944009600015625465001
524883220736003888000288000093750011250010140432
62985984298598406998400069120000337500008100000765625214565449
735831808418037760117573120014515200009450000003402000006431250050000004435633268
842998169657330892801881169920027869184000226800000001088640000030870000004800000003164062590008994801
95159780352773967052802902376448005016453120004898880000002939328000001111320000002592000000034171875001953125001798924742432
1061917364224103195607040043535646720008599633920000979776000000070543872000003333960000000103680000000020503125000023437500000118164062535499629617249
1174300837068813621820129280638522818560001418939596800001847577600000001551965184000008801654400000034214400000000902137500000015468750000001559765625007031250000693027550248468
1289161004482561783220089651209194728587264002270303354880000332563968000000031926140928000002112397056000000985374720000000324769500000000742500000000001123031250000010125000000004125976562513404343444085401
131069932053790722318186116546560130397969055744003541673233612800057644421120000000622559748096000004707627724800000025619742720000000101328084000000002895750000000000583976250000000789750000000006436523437500239257812500257176309892478032
141283918464548864299580975061401601825571566780416005409100938608640009684262748160000001162111529779200000988601822208000000614873825280000000283718635200000000972972000000000002452700250000000044226000000000005406679687500004019531250000013732910156254899270392865060249
Technical details
Node id
n-dd2a1b677086e3ba · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-399776af7ffddb1e

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:         12,          20;
  2:        144,         480,          225;
  3:       1728,        8640,         8100,         2000;
  4:      20736,      138240,       194400,        96000,        15625;
  5:     248832,     2073600,      3888000,      2880000,       937500,       112500;
  6:    2985984,    29859840,     69984000,     69120000,     33750000,      8100000,       765625;
  7:   35831808,   418037760,   1175731200,   1451520000,    945000000,    340200000,     64312500,     5000000;
  8:  429981696,  5733089280,  18811699200,  27869184000,  22680000000,  10886400000,   3087000000,   480000000,   31640625;
  9: 5159780352, 77396705280, 290237644800, 501645312000, 489888000000, 293932800000, 111132000000, 25920000000, 3417187500, 195312500;
%F Formula
The row polynomials satisfy (144+(180*n-60)*x+25*n^2*x^2)*P(n,x) = (1728+(2160*n+2160)*x+(300*n^2+1500*n+900)*x^2+(125*n^2+250*n+125)*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[((1728 + (2160*n + 2160)*x + (300*n^2 + 1500*n + 900)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(144 + (180*n - 60)*x + 25*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] = ((1728 + (2160*n + 2160)*x + (300*n^2 + 1500*n + 900)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(144 + (180*n - 60)*x + 25*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.