{"id":"n-ff2ca64442a7da44","revision":0,"first_index":0,"fingerprint":"f6-a1fb2ed15211d00c","name":"shift_x_plus_one of shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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":true,"rows":[["1"],["8","3"],["45","33","6"],["216","234","84","10"],["945","1350","720","170","15"],["3888","6885","4860","1710","300","21"],["15309","32319","28350","13230","3465","483","28"],["58320","142884","149688","86940","30240","6300","728","36"],["216513","603612","734832","510300","221130","61236","10584","1044","45"],["787320","2460375","3411720","2755620","1428840","493290","113400","16740","1440","55"],["2814669","9743085","15155910","13953060","8419950","3480246","997920","196020","25245","1925","66"],["9920232","37673262","64953900","67119030","46189440","22228668","7634088","1871100","320760","36630","2508","78"],["34543665","142780482","270208224","309613590","239246865","131351220","52540488","15428556","3301155","501930","51480","3198","91"],["119042784","531972441","1096185636","1379187810","1182160980","728999271","332756424","113837724","29189160","5540535","756756","70434","4004","105"],["406552365","1953045675","4352501790","5964539490","5615264655","3842023185","1970268300","769342860","229864635","52297245","8918910","1105650","94185","4935","120"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","84","544","3200","17664","93184","475136","2359296","11468800","54788096","257949696","1199570944","5519704064","25165824000"],"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"],["8","3"],["45","33","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["27","27","9","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-27","-18","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["9","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177353,"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-4c20296cdeac0aca"]},{"id":177354,"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-4c20296cdeac0aca","e-cd198aa968243143"]},{"id":177355,"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-4c20296cdeac0aca","e-cd198aa968243143"]},{"id":177356,"position":3,"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-cd198aa968243143"]}],"region_exact":{},"incoming_edges":[{"id":"e-cd198aa968243143","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-ff2ca64442a7da44","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":"0359ca3b2d75a2ee7d876ec45c953d5ce79391cb710b5c5ba7a85bc43e686668","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-ff3909e492eb580d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= 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]"}]},{"id":"e-4c20296cdeac0aca","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-ff2ca64442a7da44","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":"7dda2e24a68de5755e87c5bfd3de4e568e1164de4cefeaed0170b3d3d6e1c22e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-ff3909e492eb580d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= 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":[],"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:      8,       3;\n  2:     45,      33,       6;\n  3:    216,     234,      84,      10;\n  4:    945,    1350,     720,     170,      15;\n  5:   3888,    6885,    4860,    1710,     300,     21;\n  6:  15309,   32319,   28350,   13230,    3465,    483,     28;\n  7:  58320,  142884,  149688,   86940,   30240,   6300,    728,    36;\n  8: 216513,  603612,  734832,  510300,  221130,  61236,  10584,  1044,   45;\n  9: 787320, 2460375, 3411720, 2755620, 1428840, 493290, 113400, 16740, 1440, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (27+27*x+9*x^2+x^3)*P(n-3,x) - (27+18*x+3*x^2)*P(n-2,x) + (9+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 8 + 3*x, P(2,x) = 45 + 33*x + 6*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 3*x;\nP[2, x_] := 45 + 33*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[(27 + 27*x + 9*x^2 + x^3)*P[n - 3, x] - (27 + 18*x + 3*x^2)*P[n - 2, x] + (9 + 3*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] = 1; v[2] = 8 + 3*x; v[3] = 45 + 33*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (27 + 27*x + 9*x^2 + x^3)*v[i - 3] - (27 + 18*x + 3*x^2)*v[i - 2] + (9 + 3*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-787407680a56b31c","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-70e46ddc2d19c133","e-4c20296cdeac0aca"],"text":"A124932 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A124932 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}