{"id":"n-674e6855d1cd3f21","revision":0,"first_index":0,"fingerprint":"f6-747352d83647270b","name":"shift_x_plus_one of basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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"],["3","1"],["13","8","1"],["73","63","15","1"],["501","544","186","24","1"],["4051","5225","2290","430","35","1"],["37633","55656","29415","7160","855","48","1"],["394353","653023","400701","118755","18515","1533","63","1"],["4596553","8379008","5817700","2018576","385630","41888","2548","80","1"],["58941091","116780049","90064836","35620284","8012466","1071126","85764","3996","99","1"],["824073141","1757211400","1484548605","656418960","169175370","26736192","2642850","162480","5985","120","1"],["12470162233","28394129951","25994697355","12663953445","3666081210","666283926","78087702","5940330","289245","8635","143","1"],["202976401213","490371506208","482301147966","255946230760","81984834855","16794002016","2269620276","205148592","12381435","489280","12078","168","1"],["3535017524403","9013522796473","9457329713910","5417272641066","1897628103085","431482596831","65892067956","6881940780","494325117","24244935","793078","16458","195","1"],["65573803186921","175679564492264","195497275570051","119976022306504","45527247686921","11354115044272","1929281677323","228131306832","18963948003","1108379272","45046001","1239784","21931","224","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","152","1256","12032","130768","1586944","21241984","310579712","4920975104","83939634176","1532861774848","29825546764288","615785559906304"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181554,"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-a41d774ed65701cc"]},{"id":181555,"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-a41d774ed65701cc"]},{"id":181556,"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-a41d774ed65701cc"]},{"id":181557,"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-a41d774ed65701cc"]}],"region_exact":{},"incoming_edges":[{"id":"e-a41d774ed65701cc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-674e6855d1cd3f21","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":"ba5ee426ed14db3fc516ab86c4cf72ed3f957c6909de94bd91ffbb825b10fca9","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-011d67b73ad91983","role":"sequence","name":"basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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-614f2e9a399b3fee","kind":"fingerprint_related","node_a_id":"n-674e6855d1cd3f21","node_b_id":"n-54d11b5bf400eddf","fingerprint":"f6-747352d83647270b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-747352d83647270b"},"other_id":"n-54d11b5bf400eddf","other":{"id":"n-54d11b5bf400eddf","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-747352d83647270b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["13","8","1"]],"card_rows":[["1"],["3","1"],["13","8","1"],["73","63","15","1"]],"row_sums_preview":["1","4","22","152","1256","12032","130768","1586944"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4a0fd05ab0621175","steps":["x \u21a6 x+1"],"edges":["e-60b9c88fd5e752c7"],"text":"A105278 \u2192 x \u21a6 x+1"},"headline":"A105278 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d2857da4e3860846","kind":"fingerprint_related","node_a_id":"n-674e6855d1cd3f21","node_b_id":"n-0c674e242339be8f","fingerprint":"f6-747352d83647270b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-747352d83647270b"},"other_id":"n-0c674e242339be8f","other":{"id":"n-0c674e242339be8f","name":"mobius_x_over_x_plus_one of Triangular array A066667 or A008297 unsigned and transposed.","first_index":0,"fingerprint":"f6-747352d83647270b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","13"]],"card_rows":[["1"],["1","3"],["1","8","13"],["1","15","63","73"]],"row_sums_preview":["1","4","22","152","1256","12032","130768","1586944"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d55979c30b97975a","steps":["x \u21a6 x/(x+1)"],"edges":["e-31bab05fa4fccafe"],"text":"A089231 \u2192 x \u21a6 x/(x+1)"},"headline":"A089231 \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:        3,         1;\n  2:       13,         8,        1;\n  3:       73,        63,       15,        1;\n  4:      501,       544,      186,       24,       1;\n  5:     4051,      5225,     2290,      430,      35,       1;\n  6:    37633,     55656,    29415,     7160,     855,      48,     1;\n  7:   394353,    653023,   400701,   118755,   18515,    1533,    63,    1;\n  8:  4596553,   8379008,  5817700,  2018576,  385630,   41888,  2548,   80,  1;\n  9: 58941091, 116780049, 90064836, 35620284, 8012466, 1071126, 85764, 3996, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+x)*P(n-1,x) + (1+x)*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[(n + 2 + x)*P[n - 1, x] + (1 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 2 + x)*v[i - 1] + (1 + 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-efe23a76953cf69d","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-31e8b29857f291e5","e-a41d774ed65701cc"],"text":"A130534 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A130534 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}