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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"71a8d3be609c67052e811c36c736a28fc69e6b5ee8368fa46134acde25c9295c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-966c4f25f43e716e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      4,       7;\n  2:     16,      68,       61;\n  3:     64,     432,      888,       547;\n  4:    256,    2368,     7728,     10456,      4921;\n  5:   1024,   12032,    54016,    115216,    116252,      44287;\n  6:   4096,   58368,   334080,    979712,   1547616,    1245564,     398581;\n  7:  16384,  274432,  1910784,   7151360,  15501440,   19421472,   13003696,    3587227;\n  8:  65536, 1261568, 10350592,  47187968, 130512640,  223882880,  232300960,  133175792,   32285041;\n  9: 262144, 5701632, 53870592, 289787904, 976733184, 2136295680, 3028519680, 2680919520, 1343864820, 290565367;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (8*n-4+(46*n-31)*x+(84*n-66)*x^2+(62*n-55)*x^3+(16*n-16)*x^4)*P(n-1,x) - (4*x+28*x^2+60*x^3+52*x^4+16*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n - 4 + (46*n - 31)*x + (84*n - 66)*x^2 + (62*n - 55)*x^3 + (16*n - 16)*x^4)*P[n - 1, x] - (4*x + 28*x^2 + 60*x^3 + 52*x^4 + 16*x^5)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n - 4 + (46*n - 31)*x + (84*n - 66)*x^2 + (62*n - 55)*x^3 + (16*n - 16)*x^4)*v[i - 1] - (4*x + 28*x^2 + 60*x^3 + 52*x^4 + 16*x^5)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-71859d6a54438fd0","e-65c6a512ab175df7"],"text":"A103327 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A103327 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}