{"id":"n-d627fd4c1ac90a67","revision":0,"first_index":0,"fingerprint":"f6-8431aba371c0cc3d","name":"mobius_x_over_x_plus_one of Triangle of 2-Eulerian numbers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","3"],["1","9","12"],["1","21","72","60"],["1","45","300","600","360"],["1","93","1080","3900","5400","2520"],["1","189","3612","21000","50400","52920","20160"],["1","381","11592","102060","378000","670320","564480","181440"],["1","765","36300","466200","2502360","6667920","9313920","6531840","1814400"],["1","1533","111960","2046300","15309000","57524040","118540800","136080000","81648000","19958400"],["1","3069","342012","8745000","88822800","452307240","1289977920","2155507200","2095632000","1097712000","239500800"],["1","6141","1038312","36690060","496584000","3335603040","12648303360","28853858880","40415760000","34029072000","15807052800","3113510400"],["1","12285","3139500","151951800","2703060360","23489706240","115222947840","344665601280","652280428800","784864080000","582226444800","242853811200","43589145600"],["1","24573","9467640","623504700","14427012600","159859659960","994478284800","3794331340800","9317179872000","15023725516800","15822859852800","10483189516800","3966612249600","653837184000"],["1","49149","28501212","2541357000","75876091200","1060147047960","8240225273280","39304955289600","121797932256000","252726233760000","354336164582400","331495452288000","198330612480000","68652904320000","10461394944000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","154","1306","12994","148282","1908274","27333706","431220034","7428550042","138737478994","2792050329706","60231133487074","1386484468239802"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","5","7","3"]]},"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":181026,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-b55321fd7ed6faed"]},{"id":181027,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b55321fd7ed6faed"]},{"id":181028,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b55321fd7ed6faed"]}],"region_exact":{},"incoming_edges":[{"id":"e-b55321fd7ed6faed","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-d627fd4c1ac90a67","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":"b9843b3ccd5a24bd23a0b56180472bc5f20068bad3ffa72eb32369475e89c990","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-d2e17233522b2755","role":"sequence","name":"Triangle of 2-Eulerian numbers.","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":[{"id":"l-08ce74183361ebb6","kind":"fingerprint_related","node_a_id":"n-bec6dabccdd13354","node_b_id":"n-d627fd4c1ac90a67","fingerprint":"f6-8431aba371c0cc3d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8431aba371c0cc3d"},"other_id":"n-bec6dabccdd13354","other":{"id":"n-bec6dabccdd13354","name":"theta_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","first_index":1,"fingerprint":"f6-8431aba371c0cc3d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","2","6"]],"card_rows":[["1"],["0","2"],["0","2","6"],["0","2","18","24"]],"row_sums_preview":["1","2","8","44","308","2612","25988","296564"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2cabfd060b04f15d","steps":["\u03b8+1"],"edges":["e-3829ab8c6f8b03f0"],"text":"A131689 \u2192 \u03b8+1"},"headline":"A131689 \u2192 \u03b8+1","chain":null}},{"id":"l-6a3ef0033a5438d9","kind":"fingerprint_related","node_a_id":"n-d627fd4c1ac90a67","node_b_id":"n-42563b0c3ad7974d","fingerprint":"f6-8431aba371c0cc3d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8431aba371c0cc3d"},"other_id":"n-42563b0c3ad7974d","other":{"id":"n-42563b0c3ad7974d","name":"reversal of theta_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","first_index":1,"fingerprint":"f6-8431aba371c0cc3d","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["6","2"]],"card_rows":[["1"],["2"],["6","2"],["24","18","2"]],"row_sums_preview":["1","2","8","44","308","2612","25988","296564"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2cabfd060b04f15d","steps":["\u03b8+1","reverse"],"edges":["e-3829ab8c6f8b03f0","e-98a213829803e918"],"text":"A131689 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A131689 \u2192 \u03b8+1 \u2192 reverse","chain":null}}],"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: 1,    3;\n  2: 1,    9,     12;\n  3: 1,   21,     72,      60;\n  4: 1,   45,    300,     600,      360;\n  5: 1,   93,   1080,    3900,     5400,     2520;\n  6: 1,  189,   3612,   21000,    50400,    52920,     20160;\n  7: 1,  381,  11592,  102060,   378000,   670320,    564480,    181440;\n  8: 1,  765,  36300,  466200,  2502360,  6667920,   9313920,   6531840,  1814400;\n  9: 1, 1533, 111960, 2046300, 15309000, 57524040, 118540800, 136080000, 81648000, 19958400;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+5*x+7*x^2+3*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[((1 + 5*x + 7*x^2 + 3*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + 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] = ((1 + 5*x + 7*x^2 + 3*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + 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-d2e17233522b2755","steps":["x \u21a6 x/(x+1)"],"edges":["e-b55321fd7ed6faed"],"text":"A144696 \u2192 x \u21a6 x/(x+1)"},"headline":"A144696 \u2192 x \u21a6 x/(x+1)","chain":null}
