Real-rooted atlas

A038324 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(441 - 210\,x + 630\,n x + 100\,n^{2} x^{2}\right)\,P_{n}(x) = \left(9261 + 13230\,x + 6300\,x^{2} + 1000\,x^{3} + 13230\,n x + 10500\,n x^{2} + 2000\,n x^{3} + 2100\,n^{2} x^{2} + 1000\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1214061
244116809003021
39261529205670016000134881
41944811481760238140013440002500005651641
54084101388962008334900070560000262500003600000226739301
685766121980184240262549350029635200001653750000453600000490000008811313861
71801088541240145138807718950890010890936000081033750000333396000007203000000640000000334130821321
8378228593615763483331202161306249200365935449600034034175000001867017600000605052000000107520000000810000000012425939037681
979428004658113616229369960583552687284001152696666240001286491815000008821658160000038118276000000101606400000001530900000000100000000000454811023868941
10166798809782013177120186324001531825804120500345808999872000045027213525000003705096427200000200120949000000071124480000000016074450000000021000000000000121000000000016427534272151101
1135027750054222173391476304084403931686230575950099852348713040000148589804632500000142646212447200000924558784380000004107438720000000012377326500000000242550000000000027951000000000014400000000000586721655367450161
1273558275113866411681332002602660809907849301051394002795865763965120000468057884592375000051352636480992000003883146894396000000207014911488000000077977156950000000020374200000000000035218260000000000362880000000000016900000000000020753807854640862121
13154472377739119461382503030592105332024589480538063914200763271353562477760001419775583263537500001752408719913852000001514427288814440000009419178472704000000042575527694700000000139053915000000000003204861660000000000495331200000000000461370000000000001960000000000000727978271558894812981
143243919932521508681865045315339068981606024422731825658979002040016163157895104000417414021479480025000057245351517185832000005565520286393067000000395605495853568000000020862008570403000000008176370202000000000002355573320100000000004854245760000000000067821390000000000005762400000000000002250000000000000025347776045099319858741
Technical details
Node id
n-932cb7b3c3be0944 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5f0a2e78ea195dbc

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:           21,             40;
  2:          441,           1680,            900;
  3:         9261,          52920,          56700,           16000;
  4:       194481,        1481760,        2381400,         1344000,          250000;
  5:      4084101,       38896200,       83349000,        70560000,        26250000,        3600000;
  6:     85766121,      980184240,     2625493500,      2963520000,      1653750000,      453600000,       49000000;
  7:   1801088541,    24014513880,    77189508900,    108909360000,     81033750000,    33339600000,     7203000000,      640000000;
  8:  37822859361,   576348333120,  2161306249200,   3659354496000,   3403417500000,  1867017600000,   605052000000,   107520000000,    8100000000;
  9: 794280046581, 13616229369960, 58355268728400, 115269666624000, 128649181500000, 88216581600000, 38118276000000, 10160640000000, 1530900000000, 100000000000;
%F Formula
The row polynomials satisfy (441+(630*n-210)*x+100*n^2*x^2)*P(n,x) = (9261+(13230*n+13230)*x+(2100*n^2+10500*n+6300)*x^2+(1000*n^2+2000*n+1000)*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[((9261 + (13230*n + 13230)*x + (2100*n^2 + 10500*n + 6300)*x^2 + (1000*n^2 + 2000*n + 1000)*x^3)*P[n - 1, x])/(441 + (630*n - 210)*x + 100*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] = ((9261 + (13230*n + 13230)*x + (2100*n^2 + 10500*n + 6300)*x^2 + (1000*n^2 + 2000*n + 1000)*x^3)*v[i - 1])/(441 + (630*n - 210)*x + 100*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

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.