Real-rooted atlas

A034870 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(25 + 60\,x + 36\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1256036121
2625300054004320129614641
315625112500337500540000486000233280466561771561
439062537500001575000037800000567000005443200032659200111974401679616214358881
597656251171875006328125002025000000425250000061236000006123600000419904000018895680005038848006046617625937424601
624414062535156250002320312500092812500000250593750000481140000000673596000000692841600000519631200000277136640000997691904002176782336021767823363138428376721
7610351562510253906250079980468750038390625000001266890625000030405375000000547296750000007505784000000078810732000000630485856000003782915136000016507266048000495217981440091424858112078364164096379749833583241

All 15 rows

Technical details
Node id
n-21ab0e9d397e6efd · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-0d4cb94e85dd3af7

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:            25,             60,              36;
  2:           625,           3000,            5400,             4320,              1296;
  3:         15625,         112500,          337500,           540000,            486000,            233280,              46656;
  4:        390625,        3750000,        15750000,         37800000,          56700000,          54432000,           32659200,           11197440,            1679616;
  5:       9765625,      117187500,       632812500,       2025000000,        4252500000,        6123600000,         6123600000,         4199040000,         1889568000,          503884800,            60466176;
  6:     244140625,     3515625000,     23203125000,      92812500000,      250593750000,      481140000000,       673596000000,       692841600000,       519631200000,       277136640000,         99769190400,        21767823360,         2176782336;
  7:    6103515625,   102539062500,    799804687500,    3839062500000,    12668906250000,    30405375000000,     54729675000000,     75057840000000,     78810732000000,     63048585600000,      37829151360000,     16507266048000,      4952179814400,       914248581120,        78364164096;
  8:  152587890625,  2929687500000,  26367187500000,  147656250000000,   575859375000000,  1658475000000000,   3648645000000000,   6254820000000000,   8444007000000000,   9006940800000000,    7565830272000000,   4952179814400000,   2476089907200000,    914248581120000,    235092492288000,    37614798766080,     2821109907456;
  9: 3814697265625, 82397460937500, 840454101562500, 5378906250000000, 24205078125000000, 81329062500000000, 211455562500000000, 434994300000000000, 717740595000000000, 956987460000000000, 1033546456800000000, 902004180480000000, 631402926336000000, 349700082278400000, 149871463833600000, 47958868426752000, 10790745396019200, 1523399350026240, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (25+60*x+36*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[(25 + 60*x + 36*x^2)*P[n - 1, 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] = (25 + 60*x + 36*x^2)*v[i - 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.