{"id":"n-ae235faa570528c4","revision":1,"first_index":0,"fingerprint":"f6-b334f8e538f872f3","name":"basis_map_A008277 of Square the entries of Pascal's triangle.","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","1"],["6","7","1"],["20","43","15","1"],["70","251","157","26","1"],["252","1431","1415","415","40","1"],["924","8065","11791","5340","905","57","1"],["3432","45207","93821","60991","15750","1736","77","1"],["12870","252859","725173","647010","235851","39270","3038","100","1"],["48620","1413943","5499519","6535327","3208212","745941","86646","4962","126","1"],["184756","7912697","41177911","63803060","40865665","12640341","2034837","174180","7680","155","1"],["705432","44341287","305656901","607887511","496650550","197793772","41997417","4954257","325380","11385","187","1"],["2704156","248891813","2255442091","5689070398","5830086581","2920344306","790998538","122296944","11021307","572825","16291","222","1"],["10400600","1399533019","16576170897","52538191641","66665569828","41272413611","13922834354","2727196186","320547942","22781902","960245","22633","260","1"],["40116600","7883731653","121499208923","480339422836","747047808871","563947615231","232721407919","56362772752","8348687776","770876106","44306262","1544816","30667","301","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","14","79","505","3554","27083","221015","1916172","17543297","168801283","1700324080","17871445473","195456623119","2219007530714"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","1","1"],["0","-1","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","2"],["0","-1","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-1"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162315,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-906bba3237df75f5"]},{"id":162316,"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-906bba3237df75f5"]},{"id":162317,"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-906bba3237df75f5"]}],"region_exact":{},"incoming_edges":[{"id":"e-906bba3237df75f5","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-ae235faa570528c4","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"a60f3a4417f0cd618ebfd71cb36c05d61007d8a26db15af589c0748d86a6dde3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b13531ad79e7c629","role":"sequence","name":"Square the entries of Pascal's triangle.","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-059e30be86462576","kind":"fingerprint_related","node_a_id":"n-ae235faa570528c4","node_b_id":"n-045ced6e64494a1f","fingerprint":"f6-b334f8e538f872f3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b334f8e538f872f3"},"other_id":"n-045ced6e64494a1f","other":{"id":"n-045ced6e64494a1f","name":"basis_map_A008277 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-b334f8e538f872f3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["36","42","6"]],"card_rows":[["1"],["4","2"],["36","42","6"],["400","860","300","20"]],"row_sums_preview":["1","6","84","1580","35350","895608","25024692","758523480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["basis A008277"],"edges":["e-62dc2e1e9139053e"],"text":"A069466 \u2192 basis A008277"},"headline":"A069466 \u2192 basis A008277","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,       1;\n  2:     6,       7,       1;\n  3:    20,      43,      15,       1;\n  4:    70,     251,     157,      26,       1;\n  5:   252,    1431,    1415,     415,      40,      1;\n  6:   924,    8065,   11791,    5340,     905,     57,     1;\n  7:  3432,   45207,   93821,   60991,   15750,   1736,    77,    1;\n  8: 12870,  252859,  725173,  647010,  235851,  39270,  3038,  100,   1;\n  9: 48620, 1413943, 5499519, 6535327, 3208212, 745941, 86646, 4962, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -((n-1)*x+(n-1)*x^2)*P(n-2,x) - ((n-1)*x+(2*n-2)*x^2)*d/dx P(n-2,x) - (n-1)*x^2*d^2/dx^2 P(n-2,x) + (4*n-2+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n - 1)*x + (n - 1)*x^2)*P[n - 2, x] - ((n - 1)*x + (2*n - 2)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-((n - 1)*x + (n - 1)*x^2)*v[i - 2] - ((n - 1)*x + (2*n - 2)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n - 2 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x))/(n)); 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-b13531ad79e7c629","steps":["basis A008277"],"edges":["e-906bba3237df75f5"],"text":"A008459 \u2192 basis A008277"},"headline":"A008459 \u2192 basis A008277","chain":null}
