{"id":"n-136eeb95ca41b995","revision":0,"first_index":0,"fingerprint":"f6-3151c955f16eaf46","name":"theta_plus_one of derivative of Triangle of coefficients in expansion of (1+3*x)^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":true,"rows":[[],["3"],["6","36"],["9","108","243"],["12","216","972","1296"],["15","360","2430","6480","6075"],["18","540","4860","19440","36450","26244"],["21","756","8505","45360","127575","183708","107163"],["24","1008","13608","90720","340200","734832","857304","419904"],["27","1296","20412","163296","765450","2204496","3857868","3779136","1594323"],["30","1620","29160","272160","1530900","5511240","12859560","18895680","15943230","5904900"],["33","1980","40095","427680","2806650","12124728","35363790","69284160","87687765","64953900","21434787"],["36","2376","53460","641520","4811400","24249456","84873096","207852480","350751060","389723400","257217444","76527504"],["39","2808","69498","926640","7818525","45034704","183891708","540416448","1139940945","1688801400","1671913386","994857552","269440587"],["42","3276","88452","1297296","12162150","78810732","367783416","1260971712","3191834646","5910804900","7802262468","6964002864","3772168218","937461924"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","42","360","2496","15360","87552","473088","2457600","12386304","60948480","294125568","1396703232","6543114240","30299652096"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["-1","3"],["1","-6"],["0","3"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","3","0"],["0","3","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177709,"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-c3edbd0e81dc379e"]},{"id":177710,"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-c3edbd0e81dc379e"]},{"id":177711,"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-c3edbd0e81dc379e"]}],"region_exact":{},"incoming_edges":[{"id":"e-c3edbd0e81dc379e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-136eeb95ca41b995","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"fa031d4968c9aa76192fcdd6102f37b01e8f7037e813d4b4a9c93719a77bc0d4","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-fe3df7f96c918e14","role":"sequence","name":"derivative of Triangle of coefficients in expansion of (1+3*x)^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-0db29889950e8b15","kind":"fingerprint_related","node_a_id":"n-136eeb95ca41b995","node_b_id":"n-7687cf10b8b35970","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-7687cf10b8b35970","other":{"id":"n-7687cf10b8b35970","name":"theta_plus_one of basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","12","27"],["1","18","81","108"],["1","24","162","432","405"]],"card_rows":[["1","12","27"],["1","18","81","108"],["1","24","162","432","405"],["1","30","270","1080","2025","1458"]],"row_sums_preview":["40","208","1024","4864","22528","102400","458752","2031616"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013610","\u03b8+1"],"edges":["e-33def10999a9e794","e-20af71dbc536dad3"],"text":"basis-map coefficients of A112227 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A112227 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-28ae86c1ebca3590","kind":"fingerprint_related","node_a_id":"n-edecf2791d92cc84","node_b_id":"n-136eeb95ca41b995","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-edecf2791d92cc84","other":{"id":"n-edecf2791d92cc84","name":"theta_plus_one of basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-6"],["1","12","27"]],"card_rows":[["1"],["-1","-6"],["1","12","27"],["-1","-18","-81","-108"]],"row_sums_preview":["1","-7","40","-208","1024","-4864","22528","-102400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013610","\u03b8+1"],"edges":["e-fe4cc1f9dd1d3096","e-71083cdd48d9ab60"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-331c0ced8c35cd57","kind":"fingerprint_related","node_a_id":"n-5d76cb5fbd68dd8d","node_b_id":"n-136eeb95ca41b995","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-5d76cb5fbd68dd8d","other":{"id":"n-5d76cb5fbd68dd8d","name":"theta_plus_one of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","6"],["1","12","27"]],"card_rows":[["1"],["1","6"],["1","12","27"],["1","18","81","108"]],"row_sums_preview":["1","7","40","208","1024","4864","22528","102400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["\u03b8+1"],"edges":["e-f9e2d8425abfb1aa"],"text":"A013610 \u2192 \u03b8+1"},"headline":"A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-6d2ff5048492af76","kind":"fingerprint_related","node_a_id":"n-e54041bba60d6610","node_b_id":"n-136eeb95ca41b995","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-e54041bba60d6610","other":{"id":"n-e54041bba60d6610","name":"theta_plus_one of basis_map_A013610 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-6"],["1","12","27"]],"card_rows":[["1"],["-1","-6"],["1","12","27"],["-1","-18","-81","-108"]],"row_sums_preview":["1","-7","40","-208","1024","-4864","22528","-102400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013610","\u03b8+1"],"edges":["e-226103a223be7e17","e-67dba90d7253fbf9"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-b36d5a761337ecfc","kind":"fingerprint_related","node_a_id":"n-136eeb95ca41b995","node_b_id":"n-c4e421ea1b8043a3","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-c4e421ea1b8043a3","other":{"id":"n-c4e421ea1b8043a3","name":"theta_plus_one of basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","6"],["2","24","54"]],"card_rows":[["1"],["1","6"],["2","24","54"],["6","108","486","648"]],"row_sums_preview":["1","7","80","1248","24576","583680","16220160","516096000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013610","\u03b8+1"],"edges":["e-77bf2cb365051314","e-f1aec17cd456de71"],"text":"A136572 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A136572 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-c44e79b83176b47d","kind":"fingerprint_related","node_a_id":"n-136eeb95ca41b995","node_b_id":"n-2b690412bb5198c6","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-2b690412bb5198c6","other":{"id":"n-2b690412bb5198c6","name":"theta_plus_one of basis_map_A013610 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","12"],["4","48","108"]],"card_rows":[["1"],["2","12"],["4","48","108"],["8","144","648","864"]],"row_sums_preview":["1","14","160","1664","16384","155648","1441792","13107200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013610","\u03b8+1"],"edges":["e-db1885975295b772","e-fc3aaf23662ab69b"],"text":"A130123 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A130123 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-e5d35658bc00ee30","kind":"fingerprint_related","node_a_id":"n-4b82b7f00d162b38","node_b_id":"n-136eeb95ca41b995","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-4b82b7f00d162b38","other":{"id":"n-4b82b7f00d162b38","name":"theta_plus_one of basis_map_A013610 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","6"],["1","12","27"]],"card_rows":[["1"],["1","6"],["1","12","27"],["1","18","81","108"]],"row_sums_preview":["1","7","40","208","1024","4864","22528","102400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013610","\u03b8+1"],"edges":["e-41e3c7da1a12c920","e-5ef677ab8ef291c1"],"text":"basis-map coefficients of A026729 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-fb4216c237cfe126","kind":"fingerprint_related","node_a_id":"n-136eeb95ca41b995","node_b_id":"n-4ac80c0f3f62aee0","fingerprint":"f6-3151c955f16eaf46","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3151c955f16eaf46"},"other_id":"n-4ac80c0f3f62aee0","other":{"id":"n-4ac80c0f3f62aee0","name":"theta_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-3151c955f16eaf46","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","24"],["16","192","432"]],"card_rows":[["1"],["4","24"],["16","192","432"],["64","1152","5184","6912"]],"row_sums_preview":["1","28","640","13312","262144","4980736","92274688","1677721600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A013611","\u03b8+1"],"edges":["e-82b80af703a32f5e","e-70c5f3e6242d61c1"],"text":"A013610 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A013610 \u2192 basis A013611 \u2192 \u03b8+1","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:  3;\n  2:  6,   36;\n  3:  9,  108,   243;\n  4: 12,  216,   972,   1296;\n  5: 15,  360,  2430,   6480,   6075;\n  6: 18,  540,  4860,  19440,  36450,   26244;\n  7: 21,  756,  8505,  45360, 127575,  183708,  107163;\n  8: 24, 1008, 13608,  90720, 340200,  734832,  857304,  419904;\n  9: 27, 1296, 20412, 163296, 765450, 2204496, 3857868, 3779136, 1594323;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(3*n^2-6*n+3)*x)*P(n,x) = (n+(3*n^2+3*n)*x+9*n^2*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (3*n^2 + 3*n)*x + 9*n^2*x^2)*P[n - 1, x])/(n - 1 + (3*n^2 - 6*n + 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n + (3*n^2 + 3*n)*x + 9*n^2*x^2)*v[i - 1])/(n - 1 + (3*n^2 - 6*n + 3)*x)); 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-6a5af5e1168bb679","steps":["D","\u03b8+1"],"edges":["e-5fa4095a1ef98485","e-c3edbd0e81dc379e"],"text":"A013610 \u2192 D \u2192 \u03b8+1"},"headline":"A013610 \u2192 D \u2192 \u03b8+1","chain":null}
