{"id":"n-e0355357b9c46242","revision":0,"first_index":0,"fingerprint":"f6-83f6dcff4e22986d","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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":false,"rows":[["1"],["1","2"],["1","19","19"],["1","121","357","238"],["1","733","4874","8282","4141"],["1","4405","59355","193390","230730","92292"],["1","26437","687410","3834535","8198740","7537767","2512589"],["1","158629","7768599","69920280","238624360","378474822","283025078","80864308"],["1","951781","86697182","1215516903","6222366045","15150247667","19071280147","12018169364","3004542341"],["1","5710693","961124619","20528272772","151721918691","532341215532","1003351867476","1046242561308","569677949379","126595099862"],["1","34264165","10617104570","340424104595","3541652000060","17215943229774","45583983779680","69943099605390","62223217940490","29824962026095","5964992405219"],["1","205584997","117056552895","5579025104920","80269348766330","526392118932759","1881354280045093","3984317532158840","5149294593103440","3993403009254385","1709469193138839","310812580570698"],["1","1233509989","1289232497654","90722337157135","1782254956922395","15471531145634269","72638357868731546","203952767754208486","359118462424043470","400627742923775590","275333111184869809","106476852327335286","17746142054555881"],["1","7401059941","14191219969107","1467614495175468","38998114235330665","441898660708139934","2672553742201298634","9678765822945804228","22299437725352515917","33557519656929814280","32917206075857602114","20308539872955972204","7161796656854567728","1101814870285318112"],["1","44406359653","156161394629714","23658578892474987","844369903208200866","12356688578059310564","94866698777872819192","434397838821562337571","1273774237816562609193","2483037644834468385174","3259779664771178074274","2852352753100672980347","1596365418644180072511","517372289135723650363","73910327019389092909"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","39","717","18031","580173","22797479","1058836077","56769771431","3451425720333","234648926460039","17641008943213197","1453239235443241511","130180013011442568333","12599081746187570997319"],"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"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","5","15","15","5"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4","5","2"]]},"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":189439,"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-86f53241d17e4b95"]},{"id":189440,"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-86f53241d17e4b95"]},{"id":189441,"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-86f53241d17e4b95"]}],"region_exact":{},"incoming_edges":[{"id":"e-86f53241d17e4b95","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-e0355357b9c46242","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":"fac08c5d525803bc5776f5084b3c4ab8984064f187dee4a238b24e8cac256fe4","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-aac530d688cc3150","role":"sequence","name":"Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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-4a371cb7d571d6ef","kind":"fingerprint_related","node_a_id":"n-21a70536fc2023cb","node_b_id":"n-e0355357b9c46242","fingerprint":"f6-83f6dcff4e22986d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-83f6dcff4e22986d"},"other_id":"n-21a70536fc2023cb","other":{"id":"n-21a70536fc2023cb","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","first_index":0,"fingerprint":"f6-83f6dcff4e22986d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["19","19","1"]],"card_rows":[["1"],["2","1"],["19","19","1"],["238","357","121","1"]],"row_sums_preview":["1","3","39","717","18031","580173","22797479","1058836077"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-aac530d688cc3150","steps":["x \u21a6 x+1"],"edges":["e-791098f6cafefe43"],"text":"A144442 \u2192 x \u21a6 x+1"},"headline":"A144442 \u2192 x \u21a6 x+1","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: 1;\n  1: 1,       2;\n  2: 1,      19,        19;\n  3: 1,     121,       357,         238;\n  4: 1,     733,      4874,        8282,         4141;\n  5: 1,    4405,     59355,      193390,       230730,        92292;\n  6: 1,   26437,    687410,     3834535,      8198740,      7537767,       2512589;\n  7: 1,  158629,   7768599,    69920280,    238624360,    378474822,     283025078,      80864308;\n  8: 1,  951781,  86697182,  1215516903,   6222366045,  15150247667,   19071280147,   12018169364,   3004542341;\n  9: 1, 5710693, 961124619, 20528272772, 151721918691, 532341215532, 1003351867476, 1046242561308, 569677949379, 126595099862;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5*x+15*x^2+15*x^3+5*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*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 >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((5*x + 15*x^2 + 15*x^3 + 5*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*D[P[n - 1, x], 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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*x + 15*x^2 + 15*x^3 + 5*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*deriv(v[i - 1], x))/(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-aac530d688cc3150","steps":["x \u21a6 x/(x+1)"],"edges":["e-86f53241d17e4b95"],"text":"A144442 \u2192 x \u21a6 x/(x+1)"},"headline":"A144442 \u2192 x \u21a6 x/(x+1)","chain":null}