{"id":"n-67d426944a35ac1a","revision":0,"first_index":1,"fingerprint":"f6-03d79bf818d8c6e6","name":"mobius_x_over_x_plus_one of (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","4"],["6","18","13"],["6","36","63","34"],["0","24","108","156","73"],["0","0","60","240","315","136"],["0","0","0","120","450","558","229"],["0","0","0","0","210","756","903","358"],["0","0","0","0","0","336","1176","1368","529"],["0","0","0","0","0","0","504","1728","1971","748"],["0","0","0","0","0","0","0","720","2430","2730","1021"],["0","0","0","0","0","0","0","0","990","3300","3663","1354"],["0","0","0","0","0","0","0","0","0","1320","4356","4788","1753"],["0","0","0","0","0","0","0","0","0","0","1716","5616","6123","2224"],["0","0","0","0","0","0","0","0","0","0","0","2184","7098","7686","2773"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","37","139","361","751","1357","2227","3409","4951","6901","9307","12217","15679","19741"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["5","13","10","2"],["-1","-3","-3","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172651,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-0b981f66fe278cd3"]},{"id":172652,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0b981f66fe278cd3"]},{"id":172653,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0b981f66fe278cd3"]}],"region_exact":{},"incoming_edges":[{"id":"e-0b981f66fe278cd3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-67d426944a35ac1a","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"e93a9d692c2929380eb9052df2a817296a687c06253090a8a32b1f8cdb6ede9e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-77dad805f3db7972","role":"sequence","name":"(I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 3,  4;\n   3: 6, 18,  13;\n   4: 6, 36,  63,  34;\n   5: 0, 24, 108, 156,  73;\n   6: 0,  0,  60, 240, 315, 136;\n   7: 0,  0,   0, 120, 450, 558,  229;\n   8: 0,  0,   0,   0, 210, 756,  903,  358;\n   9: 0,  0,   0,   0,   0, 336, 1176, 1368,  529;\n  10: 0,  0,   0,   0,   0,   0,  504, 1728, 1971, 748;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(n-5+(3*n-13)*x+(3*n-10)*x^2+(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 5 + (3*n - 13)*x + (3*n - 10)*x^2 + (n - 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - 5 + (3*n - 13)*x + (3*n - 10)*x^2 + (n - 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["D+1","(1/n)D\u00b2+(2/n)D+1/n","x \u21a6 x/(x+1)"],"edges":["e-9e777fb0c714fbf5","e-23b6014ad5393438","e-0b981f66fe278cd3"],"text":"A132440 \u2192 D+1 \u2192 (1/n)D\u00b2+(2/n)D+1/n \u2192 x \u21a6 x/(x+1)"},"headline":"A132440 \u2192 D+1 \u2192 (1/n)D\u00b2+(2/n)D+1/n \u2192 x \u21a6 x/(x+1)","chain":null}
