{"id":"n-ef5f3a4b674fad08","revision":0,"first_index":0,"fingerprint":"f6-74fe1f9c17c9cb0a","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","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":true,"rows":[["1"],["15","19"],["225","570","361"],["3375","12825","16245","6859"],["50625","256500","487350","411540","130321"],["759375","4809375","12183750","15432750","9774075","2476099"],["11390625","86568750","274134375","462982500","439833375","222848910","47045881"],["170859375","1514953125","5756821875","12153290625","15394168125","11699567775","4939817505","893871739"],["2562890625","25970625000","115136437500","291678975000","461825043750","467982711000","296389050300","107264608680","16983563041"],["38443359375","438254296875","2220488437500","6562776937500","12469276181250","15794416496250","13337507263500","7240361085900","2292781010535","322687697779"],["576650390625","7304238281250","41634158203125","140630934375000","311731904531250","473832494887500","500156522381250","362018054295000","171958575790125","48403154666850","6131066257801"],["8649755859375","120519931640625","763292900390625","2900513021484375","7347966321093750","13030393609406250","16505165238581250","14933244739668750","9457721668456875","3993260260015125","1011625932537165","116490258898219"],["129746337890625","1972144335937500","13739272207031250","58010260429687500","165329242224609375","335067264241875000","495154957157437500","537596810628075000","425597475080559375","239595615600907500","91046333928344850","20968246601679420","2213314919066161"],["1946195068359375","32047345458984375","243559825488281250","1131200078378906250","3582133581533203125","8167264565895703125","13793602377957187500","17471896345412437500","16598301528141815625","11680286260544240625","5918011705342415250","2044404043663743450","431596409217901395","42052983462257059"],["29192926025390625","517687888183593750","4262296946044921875","21595637859960937500","75224805212197265625","190569506537566406250","362082062421376171875","524156890362373125000","580940553484963546875","490572022942858106250","310695614530476800625","143108283056462041500","45317622967879646475","8831126527073982390","799006685782884121"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","34","1156","39304","1336336","45435424","1544804416","52523350144","1785793904896","60716992766464","2064377754059776","70188843638032384","2386420683693101056","81138303245565435904","2758702310349224820736"],"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","1"]]},"terms":[{"coeff":{"coeffs":[["15","34","19"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190490,"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-39ba0cf74ce76b6c"]},{"id":190491,"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-39ba0cf74ce76b6c"]},{"id":190492,"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-39ba0cf74ce76b6c"]}],"region_exact":{},"incoming_edges":[{"id":"e-39ba0cf74ce76b6c","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-ef5f3a4b674fad08","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":"3f75bf119d4935390faeeaee3a9dd4f9bcb2176ec1fa2f690ae3c53e850a735d","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-60783defa9dbe6a9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","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:          15,           19;\n  2:         225,          570,           361;\n  3:        3375,        12825,         16245,          6859;\n  4:       50625,       256500,        487350,        411540,         130321;\n  5:      759375,      4809375,      12183750,      15432750,        9774075,        2476099;\n  6:    11390625,     86568750,     274134375,     462982500,      439833375,      222848910,       47045881;\n  7:   170859375,   1514953125,    5756821875,   12153290625,    15394168125,    11699567775,     4939817505,     893871739;\n  8:  2562890625,  25970625000,  115136437500,  291678975000,   461825043750,   467982711000,   296389050300,  107264608680,   16983563041;\n  9: 38443359375, 438254296875, 2220488437500, 6562776937500, 12469276181250, 15794416496250, 13337507263500, 7240361085900, 2292781010535, 322687697779;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (15+34*x+19*x^2)*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[((15 + 34*x + 19*x^2)*P[n - 1, x])/(1 + x)]];\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] = ((15 + 34*x + 19*x^2)*v[i - 1])/(1 + x)); 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-a679863398df5985","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-8d978ffb07d0025e","e-39ba0cf74ce76b6c"],"text":"A147716 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A147716 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}