{"id":"n-8fc0bee0ec2b787b","revision":0,"first_index":0,"fingerprint":"f6-73b18cc810b5ad72","name":"basis_map_A008277 of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["10","2"],["75","33","3"],["500","364","72","4"],["3125","3355","1085","130","5"],["18750","27906","13140","2550","210","6"],["109375","217217","139867","39130","5180","315","7"],["625000","1614488","1367184","517608","98000","9520","448","8"],["3515625","11600919","12582873","6196554","1574559","217350","16254","612","9"],["19531250","81245710","110756700","69061450","22630650","4164510","440160","26220","810","10"],["107421875","557706061","942196871","729573130","300011030","70703325","9906897","830280","40425","1045","11"],["585937500","3767627172","7803364536","7395034812","3741460800","1098593760","196013664","21676644","1479060","60060","1320","12"],["3173828125","25124342243","63257110501","72543946618","44490530513","15954681600","3525972736","494144508","44313126","2513225","86515","1638","13"],["17089843750","165759872474","503918278500","693117043710","509340161330","219732527014","58951180288","10183068896","1152539388","85613528","4104100","121394","2002","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","12","111","940","7700","62562","511091","4232256","35704755","307857470","2718390950","24611249340","228611471361","2179334356388"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196846,"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-c9d2a137ac16a4ae"]},{"id":196847,"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-c9d2a137ac16a4ae"]},{"id":196848,"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-c9d2a137ac16a4ae"]}],"region_exact":{},"incoming_edges":[{"id":"e-c9d2a137ac16a4ae","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-8fc0bee0ec2b787b","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":"7757500a04eb994b7c6ff1427443e58de6f8ba241cb9e3bfdfeeb80c3329b85c","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-ef5c9478e7d83dab","role":"sequence","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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-2185a3154182bb4f","kind":"fingerprint_related","node_a_id":"n-bac12d13055658fe","node_b_id":"n-8fc0bee0ec2b787b","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-bac12d13055658fe","other":{"id":"n-bac12d13055658fe","name":"5-Stirling numbers of the second kind.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","1"],["25","11","1"]],"card_rows":[["1"],["5","1"],["25","11","1"],["125","91","18","1"]],"row_sums_preview":["1","6","37","235","1540","10427","73013","529032"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A193685"],"derivation":{"root":"n-bac12d13055658fe","steps":[],"edges":[],"text":"A193685"},"headline":"A193685: 5-Stirling numbers of the second kind.","chain":null}},{"id":"l-26ccaf7590f4279c","kind":"fingerprint_related","node_a_id":"n-8fc0bee0ec2b787b","node_b_id":"n-19d88b15847dfa77","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-19d88b15847dfa77","other":{"id":"n-19d88b15847dfa77","name":"basis_map_A008277 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["400","176","16"],["8000","5824","1152","64"],["160000","171776","55552","6656","256"]],"card_rows":[["400","176","16"],["8000","5824","1152","64"],["160000","171776","55552","6656","256"],["3200000","4762624","2242560","435200","35840","1024"]],"row_sums_preview":["592","15040","394240","10677248","299061248","8667660288","259994091520","8070298861568"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","basis A008277"],"edges":["e-7b68ca495e98f7ea","e-5a0ab736f64a7d9b"],"text":"A038330 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A038330 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-462236661d51047c","kind":"fingerprint_related","node_a_id":"n-67748ed3490e4da5","node_b_id":"n-8fc0bee0ec2b787b","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-67748ed3490e4da5","other":{"id":"n-67748ed3490e4da5","name":"one_plus_derivative of one_plus_derivative of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","1"],["25","11","1"]],"card_rows":[["1"],["5","1"],["25","11","1"],["125","91","18","1"]],"row_sums_preview":["1","6","37","235","1540","10427","73013","529032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["D+1","D+1"],"edges":["e-71c634e76c593d53","e-df31745c2eccb007"],"text":"A143495 \u2192 D+1 \u2192 D+1"},"headline":"A143495 \u2192 D+1 \u2192 D+1","chain":null}},{"id":"l-5c5502ba5e24314f","kind":"fingerprint_related","node_a_id":"n-802eabe88630cab9","node_b_id":"n-8fc0bee0ec2b787b","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-802eabe88630cab9","other":{"id":"n-802eabe88630cab9","name":"reversal of 5-Stirling numbers of the second kind.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","5"],["1","11","25"]],"card_rows":[["1"],["1","5"],["1","11","25"],["1","18","91","125"]],"row_sums_preview":["1","6","37","235","1540","10427","73013","529032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bac12d13055658fe","steps":["reverse"],"edges":["e-1fda128e00cfeef7"],"text":"A193685 \u2192 reverse"},"headline":"A193685 \u2192 reverse","chain":null}},{"id":"l-802f84ad1b29ea36","kind":"fingerprint_related","node_a_id":"n-f0b949b616a94e3a","node_b_id":"n-8fc0bee0ec2b787b","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-f0b949b616a94e3a","other":{"id":"n-f0b949b616a94e3a","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["225","99","9"],["3375","2457","486","27"],["50625","54351","17577","2106","81"]],"card_rows":[["225","99","9"],["3375","2457","486","27"],["50625","54351","17577","2106","81"],["759375","1130193","532170","103275","8505","243"]],"row_sums_preview":["333","6345","124740","2533761","53226477","1156992984","26028766395","605955858201"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A008277"],"edges":["e-f14e261f691e9869"],"text":"A038329 \u2192 basis A008277"},"headline":"A038329 \u2192 basis A008277","chain":null}},{"id":"l-cff201f312825e1c","kind":"fingerprint_related","node_a_id":"n-8fc0bee0ec2b787b","node_b_id":"n-05ff6d2347b5ee68","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-05ff6d2347b5ee68","other":{"id":"n-05ff6d2347b5ee68","name":"basis_map_A008277 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","2"],["100","44","4"]],"card_rows":[["1"],["10","2"],["100","44","4"],["1000","728","144","8"]],"row_sums_preview":["1","12","148","1880","24640","333664","4672832","67716096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013609","basis A008277"],"edges":["e-13f16c9d95bb60f6","e-db9683720c75da67"],"text":"A027466 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A027466 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-d53f0402e7393cdb","kind":"fingerprint_related","node_a_id":"n-8fc0bee0ec2b787b","node_b_id":"n-b1ad4f9d2fc1e443","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-b1ad4f9d2fc1e443","other":{"id":"n-b1ad4f9d2fc1e443","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["20","4"],["400","176","16"]],"card_rows":[["1"],["20","4"],["400","176","16"],["8000","5824","1152","64"]],"row_sums_preview":["1","24","592","15040","394240","10677248","299061248","8667660288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","basis A008277"],"edges":["e-9c08db68889738c4","e-f32fbf019a97b46e"],"text":"A027467 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A027467 \u2192 basis A013611 \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:       0;\n  1:       1;\n  2:      10,        2;\n  3:      75,       33,        3;\n  4:     500,      364,       72,       4;\n  5:    3125,     3355,     1085,     130,       5;\n  6:   18750,    27906,    13140,    2550,     210,      6;\n  7:  109375,   217217,   139867,   39130,    5180,    315,     7;\n  8:  625000,  1614488,  1367184,  517608,   98000,   9520,   448,   8;\n  9: 3515625, 11600919, 12582873, 6196554, 1574559, 217350, 16254, 612, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (5*n+n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n + n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + n*x)*v[i - 1] + n*x*deriv(v[i - 1], x))/(n - 1)); 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-3dcaebc50e901d48","steps":["D","basis A008277"],"edges":["e-62aaaa59f91d42c1","e-c9d2a137ac16a4ae"],"text":"A038231 \u2192 D \u2192 basis A008277"},"headline":"A038231 \u2192 D \u2192 basis A008277","chain":null}