{"id":"n-78c39f1a29a465e8","revision":0,"first_index":0,"fingerprint":"f6-83d381845d6677d6","name":"mobius_x_over_x_plus_one of basis_map_A013611 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1","10","25"],["1","20","150","500","625"],["1","30","375","2500","9375","18750","15625"],["1","40","700","7000","43750","175000","437500","625000","390625"],["1","50","1125","15000","131250","787500","3281250","9375000","17578125","19531250","9765625"],["1","60","1650","27500","309375","2475000","14437500","61875000","193359375","429687500","644531250","585937500","244140625"],["1","70","2275","45500","625625","6256250","46921875","268125000","1173046875","3910156250","9775390625","17773437500","22216796875","17089843750","6103515625"],["1","80","3000","70000","1137500","13650000","125125000","893750000","5027343750","22343750000","78203125000","213281250000","444335937500","683593750000","732421875000","488281250000","152587890625"],["1","90","3825","102000","1912500","26775000","290062500","2486250000","17092968750","94960937500","427324218750","1553906250000","4532226562500","10458984375000","18676757812500","24902343750000","23345947265625","13732910156250","3814697265625"],["1","100","4750","142500","3028125","48450000","605625000","6056250000","49207031250","328046875000","1804257812500","8201171875000","30754394531250","94628906250000","236572265625000","473144531250000","739288330078125","869750976562500","724792480468750","381469726562500","95367431640625"],["1","110","5775","192500","4571875","82293750","1165828125","13323750000","124910156250","971523437500","6314902343750","34444921875000","157872558593750","607202148437500","1951721191406250","5204589843750000","11385040283203125","20091247558593750","27904510498046875","29373168945312500","22029876708984375","10490417480468750","2384185791015625"],["1","120","6900","253000","6641250","132825000","2103062500","27039375000","287293359375","2553718750000","19152890625000","121882031250000","660194335937500","3047050781250000","11970556640625000","39901855468750000","112223968505859375","264056396484375000","513442993164062500","810699462890625000","1013374328613281250","965118408203125000","658035278320312500","286102294921875000","59604644775390625"],["1","130","8125","325000","9343750","205562500","3597343750","51390625000","610263671875","6102636718750","51872412109375","377253906250000","2357836914062500","12696044921875000","58945922851562500","235783691406250000","810506439208984375","2383842468261718750","5959606170654296875","12546539306640625000","21956443786621093750","31366348266601562500","35643577575683593750","30994415283203125000","19371509552001953125","7748603820800781250","1490116119384765625"],["1","140","9450","409500","12796875","307125000","5886562500","92503125000","1214103515625","13490039062500","128155371093750","1048543945312500","7427186279296875","45705761718750000","244852294921875000","1142644042968750000","4641991424560546875","16383499145507812500","50060691833496093750","131738662719726562500","296411991119384765625","564594268798828125000","898218154907226562500","1171588897705078125000","1220405101776123046875","976324081420898437500","563263893127441406250","208616256713867187500","37252902984619140625"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","36","1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416","3656158440062976","131621703842267136","4738381338321616896","170581728179578208256","6140942214464815497216"],"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","12","46","60","25"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191866,"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-9a75854d16e0c221"]},{"id":191867,"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-9a75854d16e0c221"]},{"id":191868,"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-9a75854d16e0c221"]}],"region_exact":{},"incoming_edges":[{"id":"e-9a75854d16e0c221","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\":2}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-78c39f1a29a465e8","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":"f50e57330a585748c1d5834b2d31c5e70faa9edc8e3ed4688327e56224acf83c","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-fea317cb49e40b71","role":"sequence","name":"basis_map_A013611 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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: 1, 10,   25;\n  2: 1, 20,  150,    500,     625;\n  3: 1, 30,  375,   2500,    9375,    18750,     15625;\n  4: 1, 40,  700,   7000,   43750,   175000,    437500,     625000,      390625;\n  5: 1, 50, 1125,  15000,  131250,   787500,   3281250,    9375000,    17578125,    19531250,      9765625;\n  6: 1, 60, 1650,  27500,  309375,  2475000,  14437500,   61875000,   193359375,   429687500,    644531250,     585937500,     244140625;\n  7: 1, 70, 2275,  45500,  625625,  6256250,  46921875,  268125000,  1173046875,  3910156250,   9775390625,   17773437500,   22216796875,    17089843750,     6103515625;\n  8: 1, 80, 3000,  70000, 1137500, 13650000, 125125000,  893750000,  5027343750, 22343750000,  78203125000,  213281250000,  444335937500,   683593750000,   732421875000,   488281250000,   152587890625;\n  9: 1, 90, 3825, 102000, 1912500, 26775000, 290062500, 2486250000, 17092968750, 94960937500, 427324218750, 1553906250000, 4532226562500, 10458984375000, 18676757812500, 24902343750000, 23345947265625, 13732910156250, 3814697265625;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+12*x+46*x^2+60*x^3+25*x^4)*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 + 12*x + 46*x^2 + 60*x^3 + 25*x^4)*P[n - 1, 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 + 12*x + 46*x^2 + 60*x^3 + 25*x^4)*v[i - 1])/(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-cdb0db4ad473ee28","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-a80b60183a85d18e","e-9a75854d16e0c221"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}