{"id":"n-50ac85330e220de7","revision":0,"first_index":0,"fingerprint":"f6-74659e42bd6f64b0","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["400","1240","961"],["8000","37200","57660","29791"],["160000","992000","2306400","2383280","923521"],["3200000","24800000","76880000","119164000","92352100","28629151"],["64000000","595200000","2306400000","4766560000","5541126000","3435498120","887503681"],["1280000000","13888000000","64579200000","166829600000","258585880000","240484868400","124250515340","27512614111"],["25600000000","317440000000","1722112000000","5338547200000","10343435200000","12825859648000","9940041227200","4402018257760","852891037441"],["512000000000","7142400000000","44282880000000","160156416000000","372363667200000","577163684160000","596402473632000","396181643198400","153520386739380","26439622160671"],["10240000000000","158720000000000","1107072000000000","4575897600000000","12412122240000000","23086547366400000","29820123681600000","26412109546560000","15352038673938000","5287924432134200","819628286980801"],["204800000000000","3491840000000000","27061760000000000","125837184000000000","390095270400000000","846506736768000000","1312085441990400000","1452666025060800000","1125816169422120000","581671687534762000","180318223135776220","25408476896404831"],["4096000000000000","76185600000000000","649482240000000000","3355658240000000000","11702858112000000000","29023088117760000000","52483417679616000000","69727969202918400000","67548970165327200000","46533735002780960000","21638186776293146400","6098034455137159440","787662783788549761"],["81920000000000000","1650688000000000000","15351398400000000000","87247114240000000000","338082567680000000000","943250363827200000000","1949384085242880000000","3021545332126464000000","3512546448597014400000","3024692775180762400000","1875309520612072688000","792744479167830727200","204792323785022937860","24417546297445042591"],["1638400000000000000","35553280000000000000","358199296000000000000","2220835635200000000000","9466311895040000000000","29345566874624000000000","68228442983500800000000","120861813285058560000000","163918834267860672000000","169382795410122694400000","131271666442845088160000","73989484722330867872000","28670925329903211300400","6836912963284611925480","756943935220796320321"],["32768000000000000000","761856000000000000000","8266137600000000000000","55520890880000000000000","258172142592000000000000","880367006238720000000000","2274281432783360000000000","4532317998189696000000000","7025092897194028800000000","8469139770506134720000000","7876299986570705289600000","5549211354174815090400000","2867092532990321130040000","1025536944492691788822000","227083180566238896096300","23465261991844685929951"],["655360000000000000000","16252928000000000000000","188940288000000000000000","1366668083200000000000000","6884590469120000000000000","25610676545126400000000000","72777005849067520000000000","161149084380078080000000000","281003715887761152000000000","387160675223137587200000000","420069332617104282112000000","355149526667188165785600000","229367402639225690403200000","109390607412553790807680000","36333308890598223375408000","7508883837390299497584320","727423121747185263828481"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2601","132651","6765201","345025251","17596287801","897410677851","45767944570401","2334165173090451","119042423827613001","6071163615208263051","309629344375621415601","15791096563156692195651","805345924720991301978201","41072642160770556400888251","2094704750199298376445300801"],"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":[["400","1240","961"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["20","51","31"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194500,"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-a41c69411ecc95b1"]},{"id":194501,"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-a41c69411ecc95b1"]},{"id":194502,"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-a41c69411ecc95b1"]}],"region_exact":{},"incoming_edges":[{"id":"e-a41c69411ecc95b1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-50ac85330e220de7","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":"5d86d22a02e9ada4d8bca03a1c54b11508a38fdecc3482a64e4002e0b77eddf7","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-c02d1829f7ed6fe0","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","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-a0f994682ca5d98e","kind":"fingerprint_related","node_a_id":"n-0996adca1233ccf5","node_b_id":"n-50ac85330e220de7","fingerprint":"f6-74659e42bd6f64b0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74659e42bd6f64b0"},"other_id":"n-0996adca1233ccf5","other":{"id":"n-0996adca1233ccf5","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-74659e42bd6f64b0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["31","20"],["961","1240","400"]],"card_rows":[["1"],["31","20"],["961","1240","400"],["29791","57660","37200","8000"]],"row_sums_preview":["1","51","2601","132651","6765201","345025251","17596287801","897410677851"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["basis A007318","basis A013609"],"edges":["e-3706f9e3ae504ed3","e-b09bc21ad1cc4a76"],"text":"A038324 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A038324 \u2192 basis A007318 \u2192 basis A013609","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:             400,             1240,               961;\n  1:            8000,            37200,             57660,              29791;\n  2:          160000,           992000,           2306400,            2383280,             923521;\n  3:         3200000,         24800000,          76880000,          119164000,           92352100,           28629151;\n  4:        64000000,        595200000,        2306400000,         4766560000,         5541126000,         3435498120,           887503681;\n  5:      1280000000,      13888000000,       64579200000,       166829600000,       258585880000,       240484868400,        124250515340,         27512614111;\n  6:     25600000000,     317440000000,     1722112000000,      5338547200000,     10343435200000,     12825859648000,       9940041227200,       4402018257760,        852891037441;\n  7:    512000000000,    7142400000000,    44282880000000,    160156416000000,    372363667200000,    577163684160000,     596402473632000,     396181643198400,     153520386739380,     26439622160671;\n  8:  10240000000000,  158720000000000,  1107072000000000,   4575897600000000,  12412122240000000,  23086547366400000,   29820123681600000,   26412109546560000,   15352038673938000,   5287924432134200,    819628286980801;\n  9: 204800000000000, 3491840000000000, 27061760000000000, 125837184000000000, 390095270400000000, 846506736768000000, 1312085441990400000, 1452666025060800000, 1125816169422120000, 581671687534762000, 180318223135776220, 25408476896404831;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (20+51*x+31*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1240*x + 961*x^2.","mathematica":"P[0, x_] := 400 + 1240*x + 961*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((20 + 51*x + 31*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] = 400 + 1240*x + 961*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((20 + 51*x + 31*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-cfc5b48d32898a3e","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-1af17fa5254d2e99","e-a41c69411ecc95b1"],"text":"A038301 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A038301 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}
