Real-rooted atlas

A038334 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 150\,x + 144\,x^{2} + 90\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(500 + 5400\,x + 12240\,x^{2} + 7776\,x^{3} + 1800\,n x + 6480\,n x^{2} + 5184\,n x^{3} + 720\,n^{2} x^{2} + 864\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 3840\,x + 5184\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0400384051849424
18000115200311040221184655424
216000030720001244160017694720829440041662720
33200000768000004147200008847360008294400002866544642495550464
4640000001843200000124416000003538944000049766400000343985356809364045824143267221504
512800000004300800000034836480000012386304000002322432000000240789749760013109664153602935341711367966113284096
62560000000098304000000092897280000003963617280000092897280000000128421199872000104877313228800469654673817608916100448256432011901730816
75120000000002211840000000023887872000000011890851840000003344302080000000577895399424000062926387937280004226892064358400160489808068608026418075402240022962460071034880

All 15 rows

Technical details
Node id
n-1fffa59090d6ab2a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dc876aec23f20348

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             400,              3840,               5184;
  1:            8000,            115200,             311040,             221184;
  2:          160000,           3072000,           12441600,           17694720,             8294400;
  3:         3200000,          76800000,          414720000,          884736000,           829440000,           286654464;
  4:        64000000,        1843200000,        12441600000,        35389440000,         49766400000,         34398535680,           9364045824;
  5:      1280000000,       43008000000,       348364800000,      1238630400000,       2322432000000,       2407897497600,        1310966415360,         293534171136;
  6:     25600000000,      983040000000,      9289728000000,     39636172800000,      92897280000000,     128421199872000,      104877313228800,       46965467381760,        8916100448256;
  7:    512000000000,    22118400000000,    238878720000000,   1189085184000000,    3344302080000000,    5778953994240000,     6292638793728000,     4226892064358400,     1604898080686080,     264180754022400;
  8:  10240000000000,   491520000000000,   5971968000000000,  33973862400000000,  111476736000000000,  231158159769600000,   314631939686400000,   281792804290560000,   160489808068608000,   52836150804480000,    7671809096810496;
  9: 204800000000000, 10813440000000000, 145981440000000000, 934281216000000000, 3503554560000000000, 8475799191552000000, 13843805346201600000, 15498604235980800000, 11769252591697920000, 5811976588492800000, 1687798001298309120, 219122084616339456;
%F Formula
The row polynomials satisfy (25+(90*n+150)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (500+(1800*n+5400)*x+(720*n^2+6480*n+12240)*x^2+(864*n^2+5184*n+7776)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 3840*x + 5184*x^2.
%t Mathematica
P[0, x_] := 400 + 3840*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((500 + (1800*n + 5400)*x + (720*n^2 + 6480*n + 12240)*x^2 + (864*n^2 + 5184*n + 7776)*x^3)*P[n - 1, x])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 3840*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((500 + (1800*n + 5400)*x + (720*n^2 + 6480*n + 12240)*x^2 + (864*n^2 + 5184*n + 7776)*x^3)*v[i - 1])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.