A160562: Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).
A160562 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A160562
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 1 | 2 | |||||||||||||
| 2 | 1 | 10 | 1 | 12 | ||||||||||||
| 3 | 1 | 91 | 35 | 1 | 128 | |||||||||||
| 4 | 1 | 820 | 966 | 84 | 1 | 1872 | ||||||||||
| 5 | 1 | 7381 | 24970 | 5082 | 165 | 1 | 37600 | |||||||||
| 6 | 1 | 66430 | 631631 | 273988 | 18447 | 286 | 1 | 990784 | ||||||||
| 7 | 1 | 597871 | 15857205 | 14057043 | 1768195 | 53053 | 455 | 1 | 32333824 | |||||||
| 8 | 1 | 5380840 | 397027996 | 704652312 | 157280838 | 8187608 | 129948 | 680 | 1 | 1272660224 | ||||||
| 9 | 1 | 48427561 | 9931080740 | 34924991284 | 13444400190 | 1147981406 | 30148820 | 282948 | 969 | 1 | 59527313920 | |||||
| 10 | 1 | 435848050 | 248325446061 | 1721255653656 | 1123921406674 | 152350150316 | 6243131986 | 93812120 | 562989 | 1330 | 1 | 3252626013184 | ||||
| 11 | 1 | 3922632451 | 6208571999575 | 84589852475205 | 92758889594250 | 19558289594910 | 1207439455950 | 27350858986 | 256515941 | 1043119 | 1771 | 1 | 204354574172160 | |||
| 12 | 1 | 35303692060 | 155218222621826 | 4151111343284620 | 7598059909609455 | 2459311930578360 | 223615557650460 | 7361382727800 | 101483965935 | 633081900 | 1824130 | 2300 | 1 | 14594815769038848 | ||
| 13 | 1 | 317733228541 | 3880490869237710 | 203559674043568206 | 619593964021650475 | 305174803509591015 | 40250341173506100 | 1879926671405460 | 36690248883015 | 330026531835 | 1437523230 | 3040830 | 2925 | 1 | 1174376539738169344 | |
| 14 | 1 | 2859599056870 | 97012589464171291 | 9978304519004079804 | 50390670759797256681 | 37545745188682163290 | 7107482461832121915 | 463233842239734600 | 12483408598596795 | 155829826875450 | 963974276265 | 3046122300 | 4868955 | 3654 | 1 | 105595092426069327872 |
b-files: A008958 (reversed triangle), A160562 ?
Compute more
?
Technical details
- Node id
n-65e47d0258ed4170· revision 0 · JSON- Definition source
oeis_catalog:A160562- Fingerprint
f6-af987ff96e7c5701
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
Matches ?
- A008958 reversed triangle 45 terms, 9 rows · tabl · nonn b-file?
- A160562 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 10, 1; 3: 1, 91, 35, 1; 4: 1, 820, 966, 84, 1; 5: 1, 7381, 24970, 5082, 165, 1; 6: 1, 66430, 631631, 273988, 18447, 286, 1; 7: 1, 597871, 15857205, 14057043, 1768195, 53053, 455, 1; 8: 1, 5380840, 397027996, 704652312, 157280838, 8187608, 129948, 680, 1; 9: 1, 48427561, 9931080740, 34924991284, 13444400190, 1147981406, 30148820, 282948, 969, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*x^2*deriv(deriv(v[i - 1], x), x)); 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
A008958 reversed triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 10, 1; 3: 1, 35, 91, 1; 4: 1, 84, 966, 820, 1; 5: 1, 165, 5082, 24970, 7381, 1; 6: 1, 286, 18447, 273988, 631631, 66430, 1; 7: 1, 455, 53053, 1768195, 14057043, 15857205, 597871, 1; 8: 1, 680, 129948, 8187608, 157280838, 704652312, 397027996, 5380840, 1; 9: 1, 969, 282948, 30148820, 1147981406, 13444400190, 34924991284, 9931080740, 48427561, 1;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A160562 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 10, 1; 3: 1, 91, 35, 1; 4: 1, 820, 966, 84, 1; 5: 1, 7381, 24970, 5082, 165, 1; 6: 1, 66430, 631631, 273988, 18447, 286, 1; 7: 1, 597871, 15857205, 14057043, 1768195, 53053, 455, 1; 8: 1, 5380840, 397027996, 704652312, 157280838, 8187608, 129948, 680, 1; 9: 1, 48427561, 9931080740, 34924991284, 13444400190, 1147981406, 30148820, 282948, 969, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers
Proof chains ? JSON
References
- [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
- [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.