A convolution triangle of numbers obtained from A036224.

OEIS matches ?

• A132167: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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.

Triangle begins:
  0:          336,            42,             1;
  1:         4536,          1113,            63,            1;
  2:        54432,         23184,          2331,           84,            1;
  3:       598752,        412272,         65205,         3990,          105,           1;
  4:      6158592,       6531840,       1518048,       139860,         6090,         126,          1;
  5:     60046272,      94618368,      30912840,      4010769,       256410,        8631,        147,        1;
  6:    560431872,    1274921856,     567362016,     99302112,      8723925,      424116,      11613,      168,       1;
  7:   5043886848,   16176101760,    9573741216,   2193126624,    253790901,    16685487,     652239,    15036,     189,     1;
  8:  44019376128,  195030291456,  150628802688,  44153890560,   6534131760,   557085438,   29117907,   950040,   18900,   210,   1;
  9: 374164697088, 2250184914432, 2232875211840, 822902685696, 152347155480, 16358573673, 1096591041, 47438118, 1326780, 23205, 231, 1;

The row polynomials satisfy (n+3)*P(n,x) = (6*n+48+2*x)*P(n-1,x) + (36*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 336 + 42*x + x^2.

P[0, x_] := 336 + 42*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 48 + 2*x)*P[n - 1, x] + (36*x + x^2)*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 336 + 42*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 48 + 2*x)*v[i - 1] + (36*x + x^2)*deriv(v[i - 1], x))/(n + 3)); 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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-36, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-fc9a4a17582197de
• 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-fc9a4a17582197de
• 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-fc9a4a17582197de
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-fc9a4a17582197de

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-fc9a4a17582197de root: theorem check

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

Used in 2 proof steps

• e-0a71e6005af9c857 obreschkoff_combination → obreschkoff_combination of A convolution triangle of numbers obtained from A036224.
• e-0a71e6005af9c857 obreschkoff_combination → obreschkoff_combination of A convolution triangle of numbers obtained from A036224.

Related sequences ?

No candidate equalities or related sequences.

References

1. [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