{"id":"n-ab1ac7b8d83baeb5","revision":0,"first_index":0,"fingerprint":"f6-d62f43983c4e064b","name":"mobius_x_over_x_plus_one of Stirling2 triangle with scaled diagonals (powers of 5).","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":false,"rows":[["25","65","41"],["125","550","755","331"],["625","4375","10000","9425","3176"],["3125","35000","120000","182875","130200","35451"],["15625","290625","1406875","3077500","3466625","1968980","447981"],["78125","2531250","16565625","48656875","76993125","68372900","32206565","6282416"],["390625","23046875","197656250","748109375","1559344375","1924973875","1408163800","565971155","96546231"],["1953125","217187500","2396093750","11388359375","30061484375","48402548125","48898427500","30376496875","10637903975","1611270851"],["9765625","2095703125","29555078125","173553906250","565178593750","1140487146875","1492335882500","1273875890000","686900669500","212952636775","28985293526"],["48828125","20527343750","371583984375","2667449609375","10517507031250","25879513312500","42119320956250","46320231048125","34184585617500","16267917590750","4521005740580","558413253581"],["244140625","202880859375","4770507812500","41546689453125","195534853906250","575029851718750","1131943000687500","1539736373068750","1458048180891875","946713412717500","402871152690150","101396912076940","11458179765541"],["1220703125","2015625000000","62616210937500","657578076171875","3652725410156250","12643806571484375","29496491460937500","48255068116718750","56289660088609375","46751514012345000","27078632637950750","10415186557600800","2394297234167305","249255304141006"],["6103515625","20084228515625","840398193359375","10588591796875000","68804479443359375","277033510542968750","754210298532421875","1453946953270312500","2031463252138515625","2072006518976484375","1531380227254751875","800104851365128750","280624345089254375","59348136998068025","5725640423174901"],["30517578125","200451660156250","11520997314453125","173488840576171875","1309675973388671875","6077325260820312500","19078667328865234375","42674528962249218750","69971116124558203125","85215594618579531250","77154184428244478125","51335816288666683125","24417801048502419375","7867760144386075500","1540156622718063475","138407987170952351"],["152587890625","2002410888671875","160986022949218750","2891216326904296875","25226969726806640625","133929824161767578125","480251998018320312500","1231406889489166015625","2332834724768432812500","3323480849809148046875","3584018566184079343750","2915460856038193871875","1763070925281879314375","769334737807315010625","229178099783869075500","41746157742797939805","3510263847256823056"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["131","1761","27601","506651","10674211","251686881","6524202561","183991725451","5605930566051","183428104316161","6409252239788881","237948848526923611","9346097294356706051","386966245108218203201","16836505067572362863361"],"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":[["25","65","41"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["5","6","-4","-5"],["0","-5","-10","-5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","15","15","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191147,"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-a1488c1114be0089"]},{"id":191148,"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-a1488c1114be0089"]},{"id":191149,"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-a1488c1114be0089"]}],"region_exact":{},"incoming_edges":[{"id":"e-a1488c1114be0089","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-ab1ac7b8d83baeb5","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":"ff56b31ad15c409d8fd6d261eb3792a1dfd38903ba116adecb9010aed3065daf","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-77f76afdd288bbc1","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 5).","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:       25,          65,           41;\n  1:      125,         550,          755,           331;\n  2:      625,        4375,        10000,          9425,           3176;\n  3:     3125,       35000,       120000,        182875,         130200,          35451;\n  4:    15625,      290625,      1406875,       3077500,        3466625,        1968980,         447981;\n  5:    78125,     2531250,     16565625,      48656875,       76993125,       68372900,       32206565,        6282416;\n  6:   390625,    23046875,    197656250,     748109375,     1559344375,     1924973875,     1408163800,      565971155,       96546231;\n  7:  1953125,   217187500,   2396093750,   11388359375,    30061484375,    48402548125,    48898427500,    30376496875,    10637903975,     1611270851;\n  8:  9765625,  2095703125,  29555078125,  173553906250,   565178593750,  1140487146875,  1492335882500,  1273875890000,   686900669500,   212952636775,   28985293526;\n  9: 48828125, 20527343750, 371583984375, 2667449609375, 10517507031250, 25879513312500, 42119320956250, 46320231048125, 34184585617500, 16267917590750, 4521005740580, 558413253581;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (5-(5*n-6)*x-(10*n+4)*x^2-(5*n+5)*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 25 + 65*x + 41*x^2.","mathematica":"P[0, x_] := 25 + 65*x + 41*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((5 - (5*n - 6)*x - (10*n + 4)*x^2 - (5*n + 5)*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*D[P[n - 1, x], x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 25 + 65*x + 41*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((5 - (5*n - 6)*x - (10*n + 4)*x^2 - (5*n + 5)*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*deriv(v[i - 1], x))/(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-77f76afdd288bbc1","steps":["x \u21a6 x/(x+1)"],"edges":["e-a1488c1114be0089"],"text":"A075500 \u2192 x \u21a6 x/(x+1)"},"headline":"A075500 \u2192 x \u21a6 x/(x+1)","chain":null}
