{"id":"n-09fbd62b1b37c2d7","revision":0,"first_index":1,"fingerprint":"f6-53105ee0eb8f88ac","name":"shift_x_plus_one of derivative of Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["5","2"],["23","20","3"],["109","152","57","4"],["544","1062","672","144","5"],["2876","7282","6612","2524","345","6"],["16113","50416","60267","33884","8745","804","7"],["95495","357128","532893","400684","156655","29028","1841","8"],["597155","2604902","4679364","4437892","2359135","682842","94178","4160","9"],["3929243","19624376","41341968","47551200","32229115","12906900","2874942","302024","9297","10"],["27132324","152895932","370327284","502269332","415996730","214799976","67426198","11860120","963459","20580","11"],["196122796","1232319948","3378921972","5291287148","5196683630","3304615320","1353152794","342004528","48368025","3067360","45177","12"],["1480531285","10272302640","31490793474","56013281328","63790638335","48378476724","24593660084","8207808760","1702697130","196113070","9760740","98448","13"]],"degrees":[-1,-1,0,1,2,3,4,5,6,7,8,9,10,11,12],"row_sums":["0","0","1","7","46","322","2427","19645","170236","1573732","15459637","160769075","1763691946","20346588710","246136162031"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["10","32","24"],["-15","-40","-26"],["7","16","9"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","14","16","6"],["-4","-13","-14","-5"],["1","3","3","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["13","18"],["-10","-12"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","6","3"],["-1","-2","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174887,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-94ba39b418bd87bf"]},{"id":174888,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-010276e8f7e896b9","e-94ba39b418bd87bf"]},{"id":174889,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-010276e8f7e896b9","e-94ba39b418bd87bf"]},{"id":174890,"position":3,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-010276e8f7e896b9"]}],"region_exact":{},"incoming_edges":[{"id":"e-010276e8f7e896b9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-09fbd62b1b37c2d7","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"b35b0096ee3fc5dbc26a1bad5013fd6b566edc29c0d9f936a041e454ea4036a1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c6b873425efe3e00","role":"sequence","name":"shift_x_plus_one of Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-94ba39b418bd87bf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-09fbd62b1b37c2d7","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":"b3968676dcea758f6306c6d881289b0ec04a85be732f5823ef036e998b357904","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-bb621e5a7fb6b6af","role":"sequence","name":"derivative of Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-46462822a29b09eb","kind":"fingerprint_related","node_a_id":"n-09fbd62b1b37c2d7","node_b_id":"n-ea1d8aa155860f3a","fingerprint":"f6-53105ee0eb8f88ac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-53105ee0eb8f88ac"},"other_id":"n-ea1d8aa155860f3a","other":{"id":"n-ea1d8aa155860f3a","name":"shift_x_plus_one of derivative of Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-53105ee0eb8f88ac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["5","2"]],"card_rows":[["1"],["5","2"],["23","20","3"],["109","152","57","4"]],"row_sums_preview":["0","1","7","46","322","2427","19645","170236"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f277d949034445ba","steps":["x \u21a6 x+1","D"],"edges":["e-310068e829fb9797","e-2d110443380543aa"],"text":"A008278 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A008278 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-d6fbda6abf28208c","kind":"fingerprint_related","node_a_id":"n-0c89f20f82728361","node_b_id":"n-09fbd62b1b37c2d7","fingerprint":"f6-53105ee0eb8f88ac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-53105ee0eb8f88ac"},"other_id":"n-0c89f20f82728361","other":{"id":"n-0c89f20f82728361","name":"derivative of Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","first_index":0,"fingerprint":"f6-53105ee0eb8f88ac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["5","2"]],"card_rows":[["1"],["5","2"],["23","20","3"],["109","152","57","4"]],"row_sums_preview":["0","1","7","46","322","2427","19645","170236"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","D"],"edges":["e-236d0fb15dc6eca0","e-1c9f283a75acfd0f"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 D"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 D","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     0;\n   2:     0;\n   3:     1;\n   4:     5,      2;\n   5:    23,     20,      3;\n   6:   109,    152,     57,      4;\n   7:   544,   1062,    672,    144,      5;\n   8:  2876,   7282,   6612,   2524,    345,     6;\n   9: 16113,  50416,  60267,  33884,   8745,   804,    7;\n  10: 95495, 357128, 532893, 400684, 156655, 29028, 1841, 8;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(n^3-7*n^2+15*n-10+(2*n^3-16*n^2+40*n-32)*x+(n^3-9*n^2+26*n-24)*x^2)*P(n-2,x) + (n^2-4*n+4+(3*n^2-13*n+14)*x+(3*n^2-14*n+16)*x^2+(n^2-5*n+6)*x^3)*d/dx P(n-2,x) + (2*n^2-10*n+13+(2*n^2-12*n+18)*x)*P(n-1,x) - (n-3+(2*n-6)*x+(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 15*n - 10 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*P[n - 2, x] + (n^2 - 4*n + 4 + (3*n^2 - 13*n + 14)*x + (3*n^2 - 14*n + 16)*x^2 + (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 10*n + 13 + (2*n^2 - 12*n + 18)*x)*P[n - 1, x] - (n - 3 + (2*n - 6)*x + (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = (-(n^3 - 7*n^2 + 15*n - 10 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*v[i - 2] + (n^2 - 4*n + 4 + (3*n^2 - 13*n + 14)*x + (3*n^2 - 14*n + 16)*x^2 + (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 10*n + 13 + (2*n^2 - 12*n + 18)*x)*v[i - 1] - (n - 3 + (2*n - 6)*x + (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4ee613bf6b082e7d","steps":["x \u21a6 x+1","D"],"edges":["e-5c12eec3e0b8a462","e-010276e8f7e896b9"],"text":"A106800 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A106800 \u2192 x \u21a6 x+1 \u2192 D","chain":null}