{"id":"n-9a90a20b68ab327b","revision":0,"first_index":0,"fingerprint":"f6-74cf33eb0a23e7a7","name":"basis_map_A013609 of derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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"],["3","4"],["11","24","12"],["50","140","120","32"],["274","900","1020","480","80"],["1764","6496","8820","5600","1680","192"],["13068","52528","81228","62720","25760","5376","448"],["109584","472496","807408","718368","362880","104832","16128","1024"],["1026576","4690800","8684160","8618400","5061840","1814400","389760","46080","2304"],["10628640","51014304","100914000","109341760","72164400","30292416","8131200","1351680","126720","5120"],["120543840","603671904","1263096912","1471863360","1067162800","506411136","160125504","33454080","4435200","337920","11264"],["1486442880","7726238208","16968178656","21030618752","16485612000","8638124352","3103132032","767450112","128494080","13911040","878592","24576"],["19802759040","106386868224","243765037152","318646520192","266889502880","151861085376","60496339904","17069724672","3397185792","466385920","42025984","2236416","53248"],["283465647360","1568627191296","3731871124800","5111091381760","4533069340800","2766590002176","1201291291200","377253493760","85967481600","14078704640","1614412800","123002880","5591040","114688"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","7","47","342","2754","24552","241128","2592720","30334320","383970240","5231113920","76349105280","1188825724800","19675048780800"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","4","-4"],["2","-4","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","4"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199488,"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-6040921066078ccc"]},{"id":199489,"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-6040921066078ccc"]},{"id":199490,"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-6040921066078ccc"]},{"id":199491,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-6040921066078ccc"]}],"region_exact":{},"incoming_edges":[{"id":"e-6040921066078ccc","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-9a90a20b68ab327b","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"2a89ce43987569a1ac62f4096278ec092b3cb9da026cef2c7a589c233df098f1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0b6da90e76b5fd69","role":"sequence","name":"derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= 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-3f010cb7976896ce","kind":"fingerprint_related","node_a_id":"n-722875f27763f6cc","node_b_id":"n-9a90a20b68ab327b","fingerprint":"f6-74cf33eb0a23e7a7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74cf33eb0a23e7a7"},"other_id":"n-722875f27763f6cc","other":{"id":"n-722875f27763f6cc","name":"derivative of basis_map_A013611 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-74cf33eb0a23e7a7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["24","32"]],"card_rows":[["4"],["24","32"],["176","384","192"],["1600","4480","3840","1024"]],"row_sums_preview":["0","4","56","752","10944","176256","3142656","61728768"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013611","D"],"edges":["e-38e97daa1f393e3a","e-bcf6bf5adb5e9ef2"],"text":"A028338 \u2192 basis A013611 \u2192 D"},"headline":"A028338 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-4b08871ae8edec30","kind":"fingerprint_related","node_a_id":"n-a8864846b1c06272","node_b_id":"n-9a90a20b68ab327b","fingerprint":"f6-74cf33eb0a23e7a7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74cf33eb0a23e7a7"},"other_id":"n-a8864846b1c06272","other":{"id":"n-a8864846b1c06272","name":"derivative of basis_map_A013609 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-74cf33eb0a23e7a7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["6","8"]],"card_rows":[["2"],["6","8"],["22","48","24"],["100","280","240","64"]],"row_sums_preview":["0","2","14","94","684","5508","49104","482256"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A013609","D"],"edges":["e-5d127e92ec7ea44a","e-1ab24cef6555e268"],"text":"A132393 \u2192 basis A013609 \u2192 D"},"headline":"A132393 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-82d76efda01de9a1","kind":"fingerprint_related","node_a_id":"n-9a90a20b68ab327b","node_b_id":"n-78c8511620304395","fingerprint":"f6-74cf33eb0a23e7a7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74cf33eb0a23e7a7"},"other_id":"n-78c8511620304395","other":{"id":"n-78c8511620304395","name":"basis_map_A013609 of theta_plus_one 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.","first_index":0,"fingerprint":"f6-74cf33eb0a23e7a7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["11","24","12"]],"card_rows":[["1"],["3","4"],["11","24","12"],["50","140","120","32"]],"row_sums_preview":["1","7","47","342","2754","24552","241128","2592720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3f4ccde57d57bff1","steps":["basis A013609"],"edges":["e-07ef65626fe0142e"],"text":"A028421 \u2192 basis A013609"},"headline":"A028421 \u2192 basis A013609","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:       3,       4;\n  3:      11,      24,      12;\n  4:      50,     140,     120,      32;\n  5:     274,     900,    1020,     480,      80;\n  6:    1764,    6496,    8820,    5600,    1680,     192;\n  7:   13068,   52528,   81228,   62720,   25760,    5376,    448;\n  8:  109584,  472496,  807408,  718368,  362880,  104832,  16128,  1024;\n  9: 1026576, 4690800, 8684160, 8618400, 5061840, 1814400, 389760, 46080, 2304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(4*n-4)*x+4*x^2)*P(n-2,x) + (2*n-1+4*x)*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[-(n^2 - 2*n + 1 + (4*n - 4)*x + 4*x^2)*P[n - 2, x] + (2*n - 1 + 4*x)*P[n - 1, x]];\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] = -(n^2 - 2*n + 1 + (4*n - 4)*x + 4*x^2)*v[i - 2] + (2*n - 1 + 4*x)*v[i - 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-b9fbfb57d9269e2f","steps":["D","basis A013609"],"edges":["e-55ee0b1ee1acb060","e-6040921066078ccc"],"text":"A132393 \u2192 D \u2192 basis A013609"},"headline":"A132393 \u2192 D \u2192 basis A013609","chain":null}