{"id":"n-a53c7a97bbddcdeb","revision":0,"first_index":0,"fingerprint":"f6-d4e4e266d9f48188","name":"mobius_x_over_x_plus_one of T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.","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"],["2","3"],["6","18","13"],["24","108","158","75"],["120","720","1590","1530","541"],["720","5400","15960","23220","16622","4683"],["5040","45360","168000","327600","354606","201978","47293"],["40320","423360","1884960","4611600","6693624","5762988","2724878","545835"],["362880","4354560","22649760","66679200","121491720","140268240","100203270","40492890","7087261"],["3628800","48988800","291513600","1003363200","2200973040","3190509000","3055965720","1864917540","657944942","102247563"],["39916800","598752000","4011638400","15807052800","40558628880","70800927120","85147194000","69655330800","37093887006","11611834938","1622632573"],["479001600","7903526400","58877280000","261355248000","768024512640","1568636435520","2272003398720","2333493835200","1665417069624","786612472668","221291822798","28091567595"],["6227020800","112086374400","919004486400","4537941408000","15028603349760","35163363755520","59597732113920","73720661414400","66049526863320","41799405407760","17735890140150","4530383962650","526858348381"],["87178291200","1699976678400","15212611814400","82710403776000","304804365465600","803925509587200","1561160240640000","2259884096870400","2438441852006160","1936946544132600","1100932640360280","424010595595860","99179581033262","10641342970443"],["1307674368000","27461161728000","266329679616000","1580978310912000","6416968804646400","18837594135360000","41242909502995200","68409726703718400","86382902367281520","82632241763351280","58943279883852000","30402806419767600","10719229945624206","2312402554523898","230283190977853"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","37","365","4501","66605","1149877","22687565","503589781","12420052205","336947795317","9972186170765","319727684645461","11039636939221805","408406422098722357"],"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","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","2","-1","-4","-2"],["1","5","9","7","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-5","-9","-7","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182656,"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-f92fd716124523a0"]},{"id":182657,"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-f92fd716124523a0"]},{"id":182658,"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-f92fd716124523a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-f92fd716124523a0","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-a53c7a97bbddcdeb","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":"72c62029d63afd76034c0f2b0da21d31e7dec7964ef928306f4fe494f3f41a8e","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-3b2b9e31bd278a34","role":"sequence","name":"T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.","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-09b230258c93c45c","kind":"fingerprint_related","node_a_id":"n-a53c7a97bbddcdeb","node_b_id":"n-9cb46fc45f534e25","fingerprint":"f6-d4e4e266d9f48188","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d4e4e266d9f48188"},"other_id":"n-9cb46fc45f534e25","other":{"id":"n-9cb46fc45f534e25","name":"shift_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).","first_index":1,"fingerprint":"f6-d4e4e266d9f48188","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["13","18","6"]],"card_rows":[["1"],["3","2"],["13","18","6"],["75","158","108","24"]],"row_sums_preview":["1","5","37","365","4501","66605","1149877","22687565"],"regions":["(-2, -1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d938c1937a93c4df","steps":["x \u21a6 x+1"],"edges":["e-a640ab33da21403a"],"text":"A019538 \u2192 x \u21a6 x+1"},"headline":"A019538 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-859caa15cf2bbf46","kind":"fingerprint_related","node_a_id":"n-a53c7a97bbddcdeb","node_b_id":"n-77c7c07dd77d9ae1","fingerprint":"f6-d4e4e266d9f48188","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d4e4e266d9f48188"},"other_id":"n-77c7c07dd77d9ae1","other":{"id":"n-77c7c07dd77d9ae1","name":"mobius_x_over_x_plus_one of shift_x_plus_one of eulerian","first_index":1,"fingerprint":"f6-d4e4e266d9f48188","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["6","18","13"]],"card_rows":[["1"],["2","3"],["6","18","13"],["24","108","158","75"]],"row_sums_preview":["1","5","37","365","4501","66605","1149877","22687565"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff270bcb802a85af","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-16a5ff84df4996fb","e-d2078359af0054d1"],"text":"A008292 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A008292 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(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:       2,        3;\n  2:       6,       18,        13;\n  3:      24,      108,       158,         75;\n  4:     120,      720,      1590,       1530,        541;\n  5:     720,     5400,     15960,      23220,      16622,       4683;\n  6:    5040,    45360,    168000,     327600,     354606,     201978,      47293;\n  7:   40320,   423360,   1884960,    4611600,    6693624,    5762988,    2724878,     545835;\n  8:  362880,  4354560,  22649760,   66679200,  121491720,  140268240,  100203270,   40492890,   7087261;\n  9: 3628800, 48988800, 291513600, 1003363200, 2200973040, 3190509000, 3055965720, 1864917540, 657944942, 102247563;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n+1+(5*n+2)*x+(9*n-1)*x^2+(7*n-4)*x^3+(2*n-2)*x^4)*P(n-1,x) - (x+5*x^2+9*x^3+7*x^4+2*x^5)*d/dx 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[((n + 1 + (5*n + 2)*x + (9*n - 1)*x^2 + (7*n - 4)*x^3 + (2*n - 2)*x^4)*P[n - 1, x] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*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; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (5*n + 2)*x + (9*n - 1)*x^2 + (7*n - 4)*x^3 + (2*n - 2)*x^4)*v[i - 1] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*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-3b2b9e31bd278a34","steps":["x \u21a6 x/(x+1)"],"edges":["e-f92fd716124523a0"],"text":"A090582 \u2192 x \u21a6 x/(x+1)"},"headline":"A090582 \u2192 x \u21a6 x/(x+1)","chain":null}
