A089503 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 5\,x + x^{2} - 12\,n x - 6\,n x^{2} + 6\,n^{2} x + 13\,n^{2} x^{2} - 12\,n^{3} x^{2} + 4\,n^{4} x^{2}\right)\,P_{n}(x) = \left(1 - 11\,x + 31\,x^{2} - 22\,x^{3} + 2\,x^{4} + 4\,n x - 11\,n x^{2} - 3\,n x^{3} + n x^{4} + 6\,n^{2} x - 52\,n^{2} x^{2} + 83\,n^{2} x^{3} - 11\,n^{2} x^{4} + 28\,n^{3} x^{2} - 64\,n^{3} x^{3} + 3\,n^{3} x^{4} + 4\,n^{4} x^{2} - 10\,n^{4} x^{3} + 17\,n^{4} x^{4} + 16\,n^{5} x^{3} - 16\,n^{5} x^{4} + 4\,n^{6} x^{4}\right)\,P_{n-1}(x) + \left(-x^{2} + 6\,x^{3} - 7\,x^{4} + x^{5} - n x^{3} - 3\,n x^{4} + n x^{5} - 6\,n^{2} x^{3} + 23\,n^{2} x^{4} - 5\,n^{2} x^{5} - 10\,n^{3} x^{4} - n^{3} x^{5} - 4\,n^{4} x^{4} + 8\,n^{4} x^{5} - 4\,n^{5} x^{5}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	16														17
3	1	48	270													319
4	1	96	1512	5376												6985
5	1	160	4860	46080	126000											177101
6	1	240	11880	211200	1485000	3421440										5129761
7	1	336	24570	698880	9009000	51891840	105945840									167570467
8	1	448	45360	1881600	38220000	396264960	1977655680	3690086400								6104154449
9	1	576	77112	4386816	128520000	2072770560	18338261760	82121195520	142910127360							245575339705
10	1	720	123120	9192960	366282000	8439137280	114866035200	900174643200	3702671481600	6094932480000						10821459376081
11	1	880	187110	17740800	921690000	28668245760	552792794400	6601280716800	46995445728000	180503769600000	283932429580800					518615326284551
12	1	1056	273240	32060160	2103948000	84831183360	2193828658560	36840971059200	396328258972800	2609568783360000	9472734683596800	14347883637964800				26865637131077977
13	1	1248	386100	54912000	4440150000	225062323200	7505203305600	168046534656000	2525621258160000	24944407488000000	153931938608448000	532921392267264000	781803084142080000			1496302225059686149
14	1	1456	530712	89945856	8783775000	546421075200	22808020435200	655381485158400	13064108276419200	179205874848000000	1647977225102208000	9627851917651968000	32014836295618176000	45698009386917888000		89181623553215581025
15	1	1680	712530	141872640	16459443000	1232483091840	62907991146000	2253683367936000	57402900002448000	1039394074118400000	13205501711674272000	114295552004874240000	637158016471616640000	2046329979164190720000	2852482569352416000000	5664570733519336923691

Technical details

Node id: n-eb1a94dcb545d122 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-8350b9842fd1133b

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 -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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,  16;
   3: 1,  48,    270;
   4: 1,  96,   1512,    5376;
   5: 1, 160,   4860,   46080,    126000;
   6: 1, 240,  11880,  211200,   1485000,    3421440;
   7: 1, 336,  24570,  698880,   9009000,   51891840,    105945840;
   8: 1, 448,  45360, 1881600,  38220000,  396264960,   1977655680,   3690086400;
   9: 1, 576,  77112, 4386816, 128520000, 2072770560,  18338261760,  82121195520,  142910127360;
  10: 1, 720, 123120, 9192960, 366282000, 8439137280, 114866035200, 900174643200, 3702671481600, 6094932480000;

%F Formula

The row polynomials satisfy (1+(6*n^2-12*n+5)*x+(4*n^4-12*n^3+13*n^2-6*n+1)*x^2)*P(n,x) = (1+(6*n^2+4*n-11)*x+(4*n^4+28*n^3-52*n^2-11*n+31)*x^2+(16*n^5-10*n^4-64*n^3+83*n^2-3*n-22)*x^3+(4*n^6-16*n^5+17*n^4+3*n^3-11*n^2+n+2)*x^4)*P(n-1,x) - (x^2+(6*n^2+n-6)*x^3+(4*n^4+10*n^3-23*n^2+3*n+7)*x^4+(4*n^5-8*n^4+n^3+5*n^2-n-1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (6*n^2 + 4*n - 11)*x + (4*n^4 + 28*n^3 - 52*n^2 - 11*n + 31)*x^2 + (16*n^5 - 10*n^4 - 64*n^3 + 83*n^2 - 3*n - 22)*x^3 + (4*n^6 - 16*n^5 + 17*n^4 + 3*n^3 - 11*n^2 + n + 2)*x^4)*P[n - 1, x] - (x^2 + (6*n^2 + n - 6)*x^3 + (4*n^4 + 10*n^3 - 23*n^2 + 3*n + 7)*x^4 + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*x^5)*D[P[n - 1, x], x])/(1 + (6*n^2 - 12*n + 5)*x + (4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + (6*n^2 + 4*n - 11)*x + (4*n^4 + 28*n^3 - 52*n^2 - 11*n + 31)*x^2 + (16*n^5 - 10*n^4 - 64*n^3 + 83*n^2 - 3*n - 22)*x^3 + (4*n^6 - 16*n^5 + 17*n^4 + 3*n^3 - 11*n^2 + n + 2)*x^4)*v[i - 1] - (x^2 + (6*n^2 + n - 6)*x^3 + (4*n^4 + 10*n^3 - 23*n^2 + 3*n + 7)*x^4 + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*x^5)*deriv(v[i - 1], x))/(1 + (6*n^2 - 12*n + 5)*x + (4*n^4 - 12*n^3 + 13*n^2 - 6*n + 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of numbers used for basis change between certain falling factorials.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.