{"id":"n-f20e10cf94d19917","revision":0,"first_index":0,"fingerprint":"f6-a670175e1740e9b4","name":"shift_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":[["41","17","1"],["331","238","33","1"],["3176","3279","781","54","1"],["35451","47055","16585","1935","80","1"],["447981","718906","341440","58820","4040","111","1"],["6282416","11770347","7064246","1657460","166635","7511","147","1"],["96546231","206398693","149660183","45203606","6177745","403886","12838","188","1"],["1611270851","3863533684","3279015711","1222490809","215922126","19083351","872634","20586","234","1"],["28985293526","76900298485","74665147195","33269765220","7313549860","833685027","51218685","1726020","31395","285","1"],["558413253581","1621540048811","1770589131905","919601214015","244264187255","34694346247","2737968387","123241305","3183015","45980","341","1"],["11458179765541","36101245109552","43744984369516","25967498371320","8137828738820","1401927972522","137574653854","7918182492","271729095","5545045","65131","402","1"],["249255304141006","846021719665773","1125533470591608","751819024943636","272599244893690","55724835502622","6631751898486","473089299826","20667247887","557782940","9214491","89713","468","1"],["5725640423174901","20810828926380589","30131842623286041","22365717892281764","9232520279745126","2199021702429552","311026029342028","26855103106832","1448882934928","49574902377","1078728651","14715064","120666","539","1"],["138407987170952351","535963184846221790","838406079283183705","684462323956105605","317386787726393840","86732973935322078","14335887367481680","1470040574119260","95734588720485","4031139388705","110793035353","1983587515","22714055","159005","615","1"],["3510263847256823056","14418063813311229091","24217395312718745145","21559850636889278210","11104843153225397395","3436185600745828658","654276339258836658","78436934455190740","6055232927753295","306862349312090","10344646681798","233154717068","3493050470","34046460","205820","696","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["59","603","7291","101107","1571299","26948763","504403371","10212209987","222020715699","5151966620843","126957434503291","3308079675272147","90804952531149059","2617265143387386043","78985688314467096651"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["41","17","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191140,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-f45bb0f8dabdab85"]},{"id":191141,"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-f45bb0f8dabdab85"]},{"id":191142,"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-f45bb0f8dabdab85"]},{"id":191143,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-f45bb0f8dabdab85"]}],"region_exact":{},"incoming_edges":[{"id":"e-f45bb0f8dabdab85","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-f20e10cf94d19917","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"e3abfb1964aee6f46b9b892210995f4990c333bfce5901aa8e83f359f80a48dd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"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:           41,            17,             1;\n  1:          331,           238,            33,            1;\n  2:         3176,          3279,           781,           54,            1;\n  3:        35451,         47055,         16585,         1935,           80,           1;\n  4:       447981,        718906,        341440,        58820,         4040,         111,          1;\n  5:      6282416,      11770347,       7064246,      1657460,       166635,        7511,        147,         1;\n  6:     96546231,     206398693,     149660183,     45203606,      6177745,      403886,      12838,       188,       1;\n  7:   1611270851,    3863533684,    3279015711,   1222490809,    215922126,    19083351,     872634,     20586,     234,     1;\n  8:  28985293526,   76900298485,   74665147195,  33269765220,   7313549860,   833685027,   51218685,   1726020,   31395,   285,   1;\n  9: 558413253581, 1621540048811, 1770589131905, 919601214015, 244264187255, 34694346247, 2737968387, 123241305, 3183015, 45980, 341, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + (5+5*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 41 + 17*x + x^2.","mathematica":"P[0, x_] := 41 + 17*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(6 + x)*P[n - 1, x] + (5 + 5*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 41 + 17*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + x)*v[i - 1] + (5 + 5*x)*deriv(v[i - 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+1"],"edges":["e-f45bb0f8dabdab85"],"text":"A075500 \u2192 x \u21a6 x+1"},"headline":"A075500 \u2192 x \u21a6 x+1","chain":null}
