A056242 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-48\,x^{2} + 36\,x^{3} - 36\,x^{4} - 20\,n x - 48\,n x^{2} - 146\,n x^{3} + 156\,n x^{4} - 16\,n^{2} + 68\,n^{2} x + 287\,n^{2} x^{2} + 160\,n^{2} x^{3} - 268\,n^{2} x^{4} + 16\,n^{3} - 84\,n^{3} x - 249\,n^{3} x^{2} - 6\,n^{3} x^{3} + 228\,n^{3} x^{4} + 36\,n^{4} x + 49\,n^{4} x^{2} - 68\,n^{4} x^{3} - 96\,n^{4} x^{4} + 9\,n^{5} x^{2} + 24\,n^{5} x^{3} + 16\,n^{5} x^{4}\right)\,P_{n}(x) = \left(240\,x^{2} - 420\,x^{3} + 180\,x^{4} - 40\,n x + 256\,n x^{2} + 712\,n x^{3} - 798\,n x^{4} - 16\,n x^{5} - 32\,n^{2} - 184\,n^{2} x - 318\,n^{2} x^{2} + 563\,n^{2} x^{3} + 1304\,n^{2} x^{4} - 76\,n^{2} x^{5} + 32\,n^{3} + 24\,n^{3} x - 474\,n^{3} x^{2} - 1199\,n^{3} x^{3} - 666\,n^{3} x^{4} + 236\,n^{3} x^{5} + 72\,n^{4} x + 278\,n^{4} x^{2} + 269\,n^{4} x^{3} - 108\,n^{4} x^{4} - 192\,n^{4} x^{5} + 18\,n^{5} x^{2} + 75\,n^{5} x^{3} + 104\,n^{5} x^{4} + 48\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(48\,x^{2} - 72\,x^{3} - 48\,x^{4} + 72\,x^{5} + 88\,n x^{2} + 208\,n x^{3} - 164\,n x^{4} - 292\,n x^{5} - 8\,n x^{6} - 32\,n^{2} x - 80\,n^{2} x^{2} + 166\,n^{2} x^{3} + 562\,n^{2} x^{4} + 324\,n^{2} x^{5} - 24\,n^{2} x^{6} - 72\,n^{3} x^{2} - 260\,n^{3} x^{3} - 268\,n^{3} x^{4} - 16\,n^{3} x^{5} + 64\,n^{3} x^{6} - 18\,n^{4} x^{3} - 66\,n^{4} x^{4} - 80\,n^{4} x^{5} - 32\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	4							6
3	4	20	9						33
4	8	64	81	16					169
5	16	176	369	224	25				810
6	32	448	1314	1360	500	36			3690
7	64	1088	4104	6032	3875	972	49		16184
8	128	2560	11808	22528	20825	9324	1715	64	68952

All 15 rows

Technical details

Node id: n-6cf59f3b80d3c027 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ce5ad7513f5aaed1

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:   2,     4;
   3:   4,    20,     9;
   4:   8,    64,    81,     16;
   5:  16,   176,   369,    224,     25;
   6:  32,   448,  1314,   1360,    500,     36;
   7:  64,  1088,  4104,   6032,   3875,    972,     49;
   8: 128,  2560, 11808,  22528,  20825,   9324,   1715,    64;
   9: 256,  5888, 32112,  75392,  91225,  59472,  19894,  2816,   81;
  10: 512, 13312, 83808, 233728, 350050, 300996, 148176, 38784, 4374, 100;

%F Formula

The row polynomials satisfy (16*n^3-16*n^2+(36*n^4-84*n^3+68*n^2-20*n)*x+(9*n^5+49*n^4-249*n^3+287*n^2-48*n-48)*x^2+(24*n^5-68*n^4-6*n^3+160*n^2-146*n+36)*x^3+(16*n^5-96*n^4+228*n^3-268*n^2+156*n-36)*x^4)*P(n,x) = (32*n^3-32*n^2+(72*n^4+24*n^3-184*n^2-40*n)*x+(18*n^5+278*n^4-474*n^3-318*n^2+256*n+240)*x^2+(75*n^5+269*n^4-1199*n^3+563*n^2+712*n-420)*x^3+(104*n^5-108*n^4-666*n^3+1304*n^2-798*n+180)*x^4+(48*n^5-192*n^4+236*n^3-76*n^2-16*n)*x^5)*P(n-1,x) - (32*n^2*x+(72*n^3+80*n^2-88*n-48)*x^2+(18*n^4+260*n^3-166*n^2-208*n+72)*x^3+(66*n^4+268*n^3-562*n^2+164*n+48)*x^4+(80*n^4+16*n^3-324*n^2+292*n-72)*x^5+(32*n^4-64*n^3+24*n^2+8*n)*x^6)*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[((32*n^3 - 32*n^2 + (72*n^4 + 24*n^3 - 184*n^2 - 40*n)*x + (18*n^5 + 278*n^4 - 474*n^3 - 318*n^2 + 256*n + 240)*x^2 + (75*n^5 + 269*n^4 - 1199*n^3 + 563*n^2 + 712*n - 420)*x^3 + (104*n^5 - 108*n^4 - 666*n^3 + 1304*n^2 - 798*n + 180)*x^4 + (48*n^5 - 192*n^4 + 236*n^3 - 76*n^2 - 16*n)*x^5)*P[n - 1, x] - (32*n^2*x + (72*n^3 + 80*n^2 - 88*n - 48)*x^2 + (18*n^4 + 260*n^3 - 166*n^2 - 208*n + 72)*x^3 + (66*n^4 + 268*n^3 - 562*n^2 + 164*n + 48)*x^4 + (80*n^4 + 16*n^3 - 324*n^2 + 292*n - 72)*x^5 + (32*n^4 - 64*n^3 + 24*n^2 + 8*n)*x^6)*D[P[n - 1, x], x])/(16*n^3 - 16*n^2 + (36*n^4 - 84*n^3 + 68*n^2 - 20*n)*x + (9*n^5 + 49*n^4 - 249*n^3 + 287*n^2 - 48*n - 48)*x^2 + (24*n^5 - 68*n^4 - 6*n^3 + 160*n^2 - 146*n + 36)*x^3 + (16*n^5 - 96*n^4 + 228*n^3 - 268*n^2 + 156*n - 36)*x^4)]];
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] = ((32*n^3 - 32*n^2 + (72*n^4 + 24*n^3 - 184*n^2 - 40*n)*x + (18*n^5 + 278*n^4 - 474*n^3 - 318*n^2 + 256*n + 240)*x^2 + (75*n^5 + 269*n^4 - 1199*n^3 + 563*n^2 + 712*n - 420)*x^3 + (104*n^5 - 108*n^4 - 666*n^3 + 1304*n^2 - 798*n + 180)*x^4 + (48*n^5 - 192*n^4 + 236*n^3 - 76*n^2 - 16*n)*x^5)*v[i - 1] - (32*n^2*x + (72*n^3 + 80*n^2 - 88*n - 48)*x^2 + (18*n^4 + 260*n^3 - 166*n^2 - 208*n + 72)*x^3 + (66*n^4 + 268*n^3 - 562*n^2 + 164*n + 48)*x^4 + (80*n^4 + 16*n^3 - 324*n^2 + 292*n - 72)*x^5 + (32*n^4 - 64*n^3 + 24*n^2 + 8*n)*x^6)*deriv(v[i - 1], x))/(16*n^3 - 16*n^2 + (36*n^4 - 84*n^3 + 68*n^2 - 20*n)*x + (9*n^5 + 49*n^4 - 249*n^3 + 287*n^2 - 48*n - 48)*x^2 + (24*n^5 - 68*n^4 - 6*n^3 + 160*n^2 - 146*n + 36)*x^3 + (16*n^5 - 96*n^4 + 228*n^3 - 268*n^2 + 156*n - 36)*x^4)); 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 reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    no recorded property covers this requirement
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).?
    
    ↑ Affine-current basis map theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.