A008955: Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

OEIS matches ?

• A008955: 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 = 0 ?
• A101686: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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.

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   5,     4;
  3: 1,  14,    49,      36;
  4: 1,  30,   273,     820,      576;
  5: 1,  55,  1023,    7645,    21076,      14400;
  6: 1,  91,  3003,   44473,   296296,     773136,      518400;
  7: 1, 140,  7462,  191620,  2475473,   15291640,    38402064,    25401600;
  8: 1, 204, 16422,  669188, 14739153,  173721912,  1017067024,  2483133696,   1625702400;
  9: 1, 285, 32946, 1999370, 68943381, 1367593305, 15088541896, 84865562640, 202759531776, 131681894400;

The row polynomials satisfy P(n,x) = (1+n^2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n^2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n^2*x)*v[i - 1]); 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)

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   5,     4;
  3: 1,  14,    49,      36;
  4: 1,  30,   273,     820,      576;
  5: 1,  55,  1023,    7645,    21076,      14400;
  6: 1,  91,  3003,   44473,   296296,     773136,      518400;
  7: 1, 140,  7462,  191620,  2475473,   15291640,    38402064,    25401600;
  8: 1, 204, 16422,  669188, 14739153,  173721912,  1017067024,  2483133696,   1625702400;
  9: 1, 285, 32946, 1999370, 68943381, 1367593305, 15088541896, 84865562640, 202759531776, 131681894400;

The row polynomials satisfy P(n,x) = (1+n^2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n^2*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n^2*x)*v[i - 1]); 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)

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-c0780e435a2392b0
• 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-b6645f2226a58079, e-c0780e435a2392b0
• 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-b6645f2226a58079, e-c0780e435a2392b0
• 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-b6645f2226a58079, e-c0780e435a2392b0
• 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-b6645f2226a58079

Proof chains

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

• parent_provers paper_theorem e-b6645f2226a58079 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-c0780e435a2392b0 root: theorem check

  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 7 proof steps

• e-0b3719baa0dde23e basis_map (A007318) → basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
• e-8c95d99ebe6d3ebc basis_map (A013609) → basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
• e-a6f64f0979a8db40 basis_map (A013610) → basis_map_A013610 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
• e-bf0617bcebb15ed7 basis_map (A013611) → basis_map_A013611 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
• e-e6241a3fc3f0241e basis_map (A008277) → basis_map_A008277 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
• e-a0abdef027fe8e86 obreschkoff_combination → obreschkoff_combination of Triangle of central factorial numbers |t(2n,2n-2k)| read by ro…
• e-a0abdef027fe8e86 obreschkoff_combination → obreschkoff_combination of Triangle of central factorial numbers |t(2n,2n-2k)| read by ro…

Related sequences ?

Same fingerprint (1)

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

• A269944: Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwis…

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003