A103633 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(4 - 4\,x + 16\,x^{2} + 12\,n - 10\,n x - 72\,n x^{2} + 13\,n^{2} + 22\,n^{2} x - 28\,n^{2} x^{2} - 56\,n^{2} x^{3} + 16\,n^{2} x^{4} + 6\,n^{3} + 32\,n^{3} x + 48\,n^{3} x^{2} - 32\,n^{3} x^{4} + n^{4} + 8\,n^{4} x + 24\,n^{4} x^{2} + 32\,n^{4} x^{3} + 16\,n^{4} x^{4}\right)\,P_{n}(x) = \left(4 + 44\,x - 212\,x^{2} - 172\,x^{3} - 32\,x^{4} + 32\,x^{5} + 8\,n + 122\,n x + 178\,n x^{2} - 444\,n x^{3} - 200\,n x^{4} - 16\,n x^{5} + n^{2} + 71\,n^{2} x + 402\,n^{2} x^{2} + 536\,n^{2} x^{3} - 120\,n^{2} x^{4} - 64\,n^{2} x^{5} - 7\,n^{3} - 38\,n^{3} x + 8\,n^{3} x^{2} + 296\,n^{3} x^{3} + 384\,n^{3} x^{4} + 32\,n^{3} x^{5} - 5\,n^{4} - 45\,n^{4} x - 144\,n^{4} x^{2} - 184\,n^{4} x^{3} - 48\,n^{4} x^{4} + 48\,n^{4} x^{5} - n^{5} - 10\,n^{5} x - 40\,n^{5} x^{2} - 80\,n^{5} x^{3} - 80\,n^{5} x^{4} - 32\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(8\,x - 56\,x^{2} - 96\,x^{3} - 24\,x^{4} + 24\,x^{5} + 16\,x^{6} + 24\,n x + 20\,n x^{2} - 196\,n x^{3} - 256\,n x^{4} - 64\,n x^{5} + 26\,n^{2} x + 134\,n^{2} x^{2} + 180\,n^{2} x^{3} - 40\,n^{2} x^{4} - 144\,n^{2} x^{5} - 32\,n^{2} x^{6} + 12\,n^{3} x + 92\,n^{3} x^{2} + 256\,n^{3} x^{3} + 304\,n^{3} x^{4} + 128\,n^{3} x^{5} + 2\,n^{4} x + 18\,n^{4} x^{2} + 64\,n^{4} x^{3} + 112\,n^{4} x^{4} + 96\,n^{4} x^{5} + 32\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	4														4
2	0	4	9													13
3	0	0	9	16												25
4	0	0	9	32	25											66
5	0	0	0	16	50	36										102
6	0	0	0	16	75	108	49									248
7	0	0	0	0	25	108	147	64								344
8	0	0	0	0	25	144	294	256	81							800
9	0	0	0	0	0	36	196	384	324	100						1040
10	0	0	0	0	0	36	245	640	810	500	121					2352
11	0	0	0	0	0	0	49	320	810	1000	605	144				2928
12	0	0	0	0	0	0	49	384	1215	2000	1815	864	169			6496
13	0	0	0	0	0	0	0	64	486	1500	2420	2160	1014	196		7840
14	0	0	0	0	0	0	0	64	567	2100	4235	5040	3549	1372	225	17152

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Technical details

Node id: n-2c235d997a804fe6 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-fd66d757952530e2

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 0, 4;
  2: 0, 4, 9;
  3: 0, 0, 9, 16;
  4: 0, 0, 9, 32, 25;
  5: 0, 0, 0, 16, 50,  36;
  6: 0, 0, 0, 16, 75, 108,  49;
  7: 0, 0, 0,  0, 25, 108, 147,  64;
  8: 0, 0, 0,  0, 25, 144, 294, 256,  81;
  9: 0, 0, 0,  0,  0,  36, 196, 384, 324, 100;

%F Formula

The row polynomials satisfy (n^4+6*n^3+13*n^2+12*n+4+(8*n^4+32*n^3+22*n^2-10*n-4)*x+(24*n^4+48*n^3-28*n^2-72*n+16)*x^2+(32*n^4-56*n^2)*x^3+(16*n^4-32*n^3+16*n^2)*x^4)*P(n,x) = -(n^5+5*n^4+7*n^3-n^2-8*n-4+(10*n^5+45*n^4+38*n^3-71*n^2-122*n-44)*x+(40*n^5+144*n^4-8*n^3-402*n^2-178*n+212)*x^2+(80*n^5+184*n^4-296*n^3-536*n^2+444*n+172)*x^3+(80*n^5+48*n^4-384*n^3+120*n^2+200*n+32)*x^4+(32*n^5-48*n^4-32*n^3+64*n^2+16*n-32)*x^5)*P(n-1,x) + ((2*n^4+12*n^3+26*n^2+24*n+8)*x+(18*n^4+92*n^3+134*n^2+20*n-56)*x^2+(64*n^4+256*n^3+180*n^2-196*n-96)*x^3+(112*n^4+304*n^3-40*n^2-256*n-24)*x^4+(96*n^4+128*n^3-144*n^2-64*n+24)*x^5+(32*n^4-32*n^2+16)*x^6)*d/dx 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[(-(n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 + (10*n^5 + 45*n^4 + 38*n^3 - 71*n^2 - 122*n - 44)*x + (40*n^5 + 144*n^4 - 8*n^3 - 402*n^2 - 178*n + 212)*x^2 + (80*n^5 + 184*n^4 - 296*n^3 - 536*n^2 + 444*n + 172)*x^3 + (80*n^5 + 48*n^4 - 384*n^3 + 120*n^2 + 200*n + 32)*x^4 + (32*n^5 - 48*n^4 - 32*n^3 + 64*n^2 + 16*n - 32)*x^5)*P[n - 1, x] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x + (18*n^4 + 92*n^3 + 134*n^2 + 20*n - 56)*x^2 + (64*n^4 + 256*n^3 + 180*n^2 - 196*n - 96)*x^3 + (112*n^4 + 304*n^3 - 40*n^2 - 256*n - 24)*x^4 + (96*n^4 + 128*n^3 - 144*n^2 - 64*n + 24)*x^5 + (32*n^4 - 32*n^2 + 16)*x^6)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (8*n^4 + 32*n^3 + 22*n^2 - 10*n - 4)*x + (24*n^4 + 48*n^3 - 28*n^2 - 72*n + 16)*x^2 + (32*n^4 - 56*n^2)*x^3 + (16*n^4 - 32*n^3 + 16*n^2)*x^4)]];
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] = (-(n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 + (10*n^5 + 45*n^4 + 38*n^3 - 71*n^2 - 122*n - 44)*x + (40*n^5 + 144*n^4 - 8*n^3 - 402*n^2 - 178*n + 212)*x^2 + (80*n^5 + 184*n^4 - 296*n^3 - 536*n^2 + 444*n + 172)*x^3 + (80*n^5 + 48*n^4 - 384*n^3 + 120*n^2 + 200*n + 32)*x^4 + (32*n^5 - 48*n^4 - 32*n^3 + 64*n^2 + 16*n - 32)*x^5)*v[i - 1] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x + (18*n^4 + 92*n^3 + 134*n^2 + 20*n - 56)*x^2 + (64*n^4 + 256*n^3 + 180*n^2 - 196*n - 96)*x^3 + (112*n^4 + 304*n^3 - 40*n^2 - 256*n - 24)*x^4 + (96*n^4 + 128*n^3 - 144*n^2 - 64*n + 24)*x^5 + (32*n^4 - 32*n^2 + 16)*x^6)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (8*n^4 + 32*n^3 + 22*n^2 - 10*n - 4)*x + (24*n^4 + 48*n^3 - 28*n^2 - 72*n + 16)*x^2 + (32*n^4 - 56*n^2)*x^3 + (16*n^4 - 32*n^3 + 16*n^2)*x^4)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-71ab500f4221c0be
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-71ab500f4221c0be
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-71ab500f4221c0be

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image euler_polynomial, shift 0 e-71ab500f4221c0be

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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.