{"id":"n-9128a95b5b9242bf","revision":0,"first_index":0,"fingerprint":"f6-67f6b8af5e681e1c","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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"],["3","5"],["18","54","40"],["162","684","954","440"],["1944","10476","21024","18636","6160"],["29160","190188","493452","636948","409212","104720"],["524880","4006584","12685896","21334104","20104776","10069200","2094400"],["11022480","96210504","358555248","739791792","912884688","673863048","275574000","48171200"],["264539520","2595636576","11106799056","27077439456","41143819104","39907715040","24134215248","8321032800","1252451200"],["7142567040","77753833632","375157035216","1053188037936","1896128860032","2270679060096","1808947547856","924560127792","275126133600","36321084800"],["214277011200","2561177154240","13742833732704","43600666264992","90585883014912","128796495323904","126930156358752","85623125365152","37839708097344","9893668819200","1162274713600"],["7071141369600","92018541481920","543154713577632","1919821167389376","4515357458766816","7420790251210752","8696572496175456","7268118609606336","4245519754992672","1650880854440640","384633474220800","40679614976000"],["254561089305600","3581370865393920","23050274265107712","89753441141967552","235506072876401664","438732032476726080","595066639408324992","592122024800495424","429027218344776960","220761461449584576","76580277540693120","16080538159526400","1545825369088000"],["9927882482918400","150110468411892480","1045796901820351488","4445445449406150720","12864627929000334528","26766298254524785344","41195612268470443968","47490491182960646208","41009064532266613824","26199812948669653824","12037850743519999296","3766932367389947520","719589253935014400","63378840132608000"],["416971064282572800","6741466502547893760","50528330486578031616","232723772555153795712","735913972791884681856","1690228155566055703680","2907932838474849201792","3807207569497046675328","3811962322405466213760","2904790983263856468864","1658381500969304738688","687876592145257764864","195967772830428295680","34323838458710169600","2788668965834752000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","112","2240","58240","1863680","70819840","3116072960","155803648000","8725004288000","540950265856000","36784618078208000","2722061737787392000","217764939022991360000","18727784755977256960000"],"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":[["0","2","2"],["3","6","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188212,"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-6babaae0bec71062"]},{"id":188213,"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-6babaae0bec71062"]},{"id":188214,"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-6babaae0bec71062"]}],"region_exact":{},"incoming_edges":[{"id":"e-6babaae0bec71062","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-9128a95b5b9242bf","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":"391b1b2ef1f1ca35b55f5cb5e31939c9578df1a2ca0eeeee052a5b7cbad4e5f7","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-a33b47c8b957d83a","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 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:          3,           5;\n  2:         18,          54,           40;\n  3:        162,         684,          954,           440;\n  4:       1944,       10476,        21024,         18636,          6160;\n  5:      29160,      190188,       493452,        636948,        409212,        104720;\n  6:     524880,     4006584,     12685896,      21334104,      20104776,      10069200,       2094400;\n  7:   11022480,    96210504,    358555248,     739791792,     912884688,     673863048,     275574000,     48171200;\n  8:  264539520,  2595636576,  11106799056,   27077439456,   41143819104,   39907715040,   24134215248,   8321032800,   1252451200;\n  9: 7142567040, 77753833632, 375157035216, 1053188037936, 1896128860032, 2270679060096, 1808947547856, 924560127792, 275126133600, 36321084800;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (3*n+(6*n+2)*x+(3*n+2)*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[((3*n + (6*n + 2)*x + (3*n + 2)*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] = ((3*n + (6*n + 2)*x + (3*n + 2)*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-8229dbdd6a4b5cd9","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-0826250557dc6011","e-6babaae0bec71062"],"text":"A225470 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A225470 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}