A269939: Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.

A269939 nonnegative coefficients

Proved for all n: all roots real, in (−1, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A269939

\(P_{n}(x) = \left(-x + n x\right)\,P_{n-1}(x) + \left(x + x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	0	1							1
3	0	1	3						4
4	0	1	10	15					26
5	0	1	25	105	105				236
6	0	1	56	490	1260	945			2752
7	0	1	119	1918	9450	17325	10395		39208
8	0	1	246	6825	56980	190575	270270	135135	660032

Show all 15 stored rows

Generated b-files: A134991, A181996 (reversed triangle), A269939 ?

Computed in your browser ?

Technical details

Node id: n-eda3c556a6b4857f · revision 1 · JSON
Definition source: oeis_catalog:A269939
Fingerprint: f6-014ae34d4ff839c2

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

OEIS matches ?

A181996: reversed triangle, 46 terms agree, 10 full rows row lengths from the atlas (tabf) OEIS nonn · view at the OEIS · generated b-file 991 terms from n = 0 ?
A000311: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A269939: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A134991: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 1,    3;
   4: 0, 1,   10,    15;
   5: 0, 1,   25,   105,     105;
   6: 0, 1,   56,   490,    1260,      945;
   7: 0, 1,  119,  1918,    9450,    17325,    10395;
   8: 0, 1,  246,  6825,   56980,   190575,   270270,   135135;
   9: 0, 1,  501, 22935,  302995,  1636635,  4099095,  4729725,  2027025;
  10: 0, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (n-1)*x*P(n-1,x) + (x+x^2)*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[(n - 1)*x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
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] = (n - 1)*x*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); 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)

Generated with polytool c48629374597.

A134991: triangle, offset 1, from the row P₂, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,    3;
   3: 1,   10,     15;
   4: 1,   25,    105,     105;
   5: 1,   56,    490,    1260,      945;
   6: 1,  119,   1918,    9450,    17325,     10395;
   7: 1,  246,   6825,   56980,   190575,    270270,     135135;
   8: 1,  501,  22935,  302995,  1636635,   4099095,    4729725,    2027025;
   9: 1, 1012,  74316, 1487200, 12122110,  47507460,   94594500,   91891800,   34459425;
  10: 1, 2035, 235092, 6914908, 81431350, 466876410, 1422280860, 2343240900, 1964187225, 654729075;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = n*x*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.

%t Mathematica

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = n*x*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A181996: reversed triangle, offset 0, from the row P₁, low zeros dropped

%e Example

Triangle begins:
  0:        1;
  1:        1;
  2:        3,        1;
  3:       15,       10,        1;
  4:      105,      105,       25,        1;
  5:      945,     1260,      490,       56,        1;
  6:    10395,    17325,     9450,     1918,      119,       1;
  7:   135135,   270270,   190575,    56980,     6825,     246,     1;
  8:  2027025,  4729725,  4099095,  1636635,   302995,   22935,   501,    1;
  9: 34459425, 91891800, 94594500, 47507460, 12122110, 1487200, 74316, 1012, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = n*x*P(n-1,x) + (x+x^2)*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[n*x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], 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*x*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A269939: triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,    3;
  3: 0, 1,   10,    15;
  4: 0, 1,   25,   105,     105;
  5: 0, 1,   56,   490,    1260,      945;
  6: 0, 1,  119,  1918,    9450,    17325,    10395;
  7: 0, 1,  246,  6825,   56980,   190575,   270270,   135135;
  8: 0, 1,  501, 22935,  302995,  1636635,  4099095,  4729725,  2027025;
  9: 0, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = n*x*P(n-1,x) + (x+x^2)*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[n*x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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*x*v[i - 1] + (x + x^2)*deriv(v[i - 1], 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-ade83b202f7a862b
Coefficients of one sign. Every nonzero row, n >= 1: 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-ade83b202f7a862b
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-ade83b202f7a862b
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-ade83b202f7a862b

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

theorem_check ma_wang_interval e-ade83b202f7a862b root: theorem check

Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 2 proof steps

e-0730b0b2fd7e59bf obreschkoff_combination → obreschkoff_combination of Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1…
e-0730b0b2fd7e59bf obreschkoff_combination → obreschkoff_combination of Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1…

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A134991: Triangle of Ward numbers T(n,k) read by rows.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741