{"id":"n-d745086d861ef817","revision":0,"first_index":0,"fingerprint":"f6-1f0181c04d99414a","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4"],["10","40"],["20","240"],["35","840","315"],["56","2240","2520"],["84","5040","11340","1344"],["120","10080","37800","13440"],["165","18480","103950","73920","4125"],["220","31680","249480","295680","49500"],["286","51480","540540","960960","321750","10296"],["364","80080","1081080","2690688","1501500","144144"],["455","120120","2027025","6726720","5630625","1081080","22295"],["560","174720","3603600","15375360","18018000","5765760","356720"],["680","247520","6126120","32672640","51051000","24504480","3032120","43520"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","4","50","260","1190","4816","17808","61440","200640","626560","1885312","5497856","15608320","43294720","117678080"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-36","-12","0"],["36","10","0"],["0","24","-1"],["0","-22","-1"],["0","0","1"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-108","-468","12","-12"],["72","306","190","-28"],["36","226","-25","-11"],["0","-42","-146","20"],["0","-22","-36","22"],["0","0","4","8"],["0","0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","216","-264","72","-24"],["0","72","-340","348","-80"],["0","0","-216","318","-102"],["0","0","-44","106","-62"],["0","0","0","18","-18"],["0","0","0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175532,"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-4168e5c5dcbea788"]},{"id":175533,"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-4168e5c5dcbea788"]},{"id":175534,"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-4168e5c5dcbea788"]}],"region_exact":{},"incoming_edges":[{"id":"e-4168e5c5dcbea788","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-d745086d861ef817","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":"75b99924b2a01995c3e3012a4d308196aaf52193428622d86dd7982ed19a4955","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-7f77cea6aed0df8c","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 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-1f8ffbd18ce9107d","kind":"fingerprint_related","node_a_id":"n-202941dcb2123fea","node_b_id":"n-d745086d861ef817","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-202941dcb2123fea","other":{"id":"n-202941dcb2123fea","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["2","8"]],"card_rows":[["1"],["2"],["2","8"],["2","24"]],"row_sums_preview":["1","2","10","26","68","172","424","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ced86a7ce48c966e","steps":["(\u03b8+1)\u00b2"],"edges":["e-84b40fb5012c9317"],"text":"A119462 \u2192 (\u03b8+1)\u00b2"},"headline":"A119462 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-422ea9f60059ffb2","kind":"fingerprint_related","node_a_id":"n-d745086d861ef817","node_b_id":"n-745a4a6b272e9e4d","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-745a4a6b272e9e4d","other":{"id":"n-745a4a6b272e9e4d","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6"],["21","84"]],"card_rows":[["1"],["6"],["21","84"],["56","672"]],"row_sums_preview":["1","6","105","728","4284","21672","97944","405504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-496b1966a8762df1","e-3a36bf9839123f38"],"text":"A178822 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A178822 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-59e63cefaa8898c9","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-d745086d861ef817","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-841b8a0af1d94710","other":{"id":"n-841b8a0af1d94710","name":"theta_plus_one_squared of veronese_section_2_0 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1"],["1","4"]],"card_rows":[["1"],["-1"],["1","4"],["-1","-12"]],"row_sums_preview":["1","-1","5","-13","34","-86","212","-512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-969b818b29eb86a6","e-01d022a27a535273"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-61d180a00f6f4047","kind":"fingerprint_related","node_a_id":"n-d745086d861ef817","node_b_id":"n-1505e8f7a40e80bf","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-1505e8f7a40e80bf","other":{"id":"n-1505e8f7a40e80bf","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5"],["15","60"]],"card_rows":[["1"],["5"],["15","60"],["35","420"]],"row_sums_preview":["1","5","75","455","2380","10836","44520","168960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-8ba085c2fb60bb1b","e-8504682fd01917c7"],"text":"A178821 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A178821 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-7c5fd40d1f7477bb","kind":"fingerprint_related","node_a_id":"n-d745086d861ef817","node_b_id":"n-2a832b7b6a769b3d","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-2a832b7b6a769b3d","other":{"id":"n-2a832b7b6a769b3d","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","16"]],"card_rows":[["1"],["2"],["4","16"],["8","96"]],"row_sums_preview":["1","2","20","104","544","2752","13568","65536"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-01a436d0e1f57933","e-46a525a2e4f7215b"],"text":"A038208 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A038208 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-7e472c511d4625fe","kind":"fingerprint_related","node_a_id":"n-d745086d861ef817","node_b_id":"n-6ec34513293cdda0","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-6ec34513293cdda0","other":{"id":"n-6ec34513293cdda0","name":"theta_plus_one_squared of Table: T(n,k) = n!*binomial(n+1,2*k).","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["2","24"]],"card_rows":[["1"],["1","4"],["2","24"],["6","144","54"]],"row_sums_preview":["1","5","26","204","2064","25440","368640","6128640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-90f6949024558b8d","steps":["(\u03b8+1)\u00b2"],"edges":["e-4ed12709a7f73fb2"],"text":"A228955 \u2192 (\u03b8+1)\u00b2"},"headline":"A228955 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-ae1b3e020f7ce786","kind":"fingerprint_related","node_a_id":"n-af0246c88c139e87","node_b_id":"n-d745086d861ef817","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-af0246c88c139e87","other":{"id":"n-af0246c88c139e87","name":"theta_plus_one_squared of veronese_section_2_0 of binomial","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","12"]],"row_sums_preview":["1","1","5","13","34","86","212","512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9cbb4c37ddc3c985","steps":["(\u03b8+1)\u00b2"],"edges":["e-1da06d7d5e63865c"],"text":"A034839 \u2192 (\u03b8+1)\u00b2"},"headline":"A034839 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-f7c3e30a78cb44d3","kind":"fingerprint_related","node_a_id":"n-094763b148064a77","node_b_id":"n-d745086d861ef817","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-094763b148064a77","other":{"id":"n-094763b148064a77","name":"theta_plus_one_squared of veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2"],["2","8"]],"card_rows":[["2"],["2"],["2","8"],["2","24"]],"row_sums_preview":["2","2","10","26","68","172","424","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-e811ab86eff6d185","e-1fe02b877a13aae0"],"text":"A028326 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A028326 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","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:   1;\n  1:   4;\n  2:  10,    40;\n  3:  20,   240;\n  4:  35,   840,    315;\n  5:  56,  2240,   2520;\n  6:  84,  5040,  11340,   1344;\n  7: 120, 10080,  37800,  13440;\n  8: 165, 18480, 103950,  73920,  4125;\n  9: 220, 31680, 249480, 295680, 49500;\n","recurrence_sentence":"The row polynomials satisfy (36*n^2-36*n-(22*n^4-24*n^3-10*n^2+12*n)*x+(n^6+n^5-n^4-n^3)*x^2)*P(n,x) = (36*n^2+72*n-108-(22*n^4+42*n^3-226*n^2-306*n+468)*x+(n^6+4*n^5-36*n^4-146*n^3-25*n^2+190*n+12)*x^2+(n^6+8*n^5+22*n^4+20*n^3-11*n^2-28*n-12)*x^3)*P(n-1,x) + ((72*n+216)*x-(44*n^3+216*n^2+340*n+264)*x^2+(2*n^5+18*n^4+106*n^3+318*n^2+348*n+72)*x^3-(2*n^5+18*n^4+62*n^3+102*n^2+80*n+24)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((36*n^2 + 72*n - 108 - (22*n^4 + 42*n^3 - 226*n^2 - 306*n + 468)*x + (n^6 + 4*n^5 - 36*n^4 - 146*n^3 - 25*n^2 + 190*n + 12)*x^2 + (n^6 + 8*n^5 + 22*n^4 + 20*n^3 - 11*n^2 - 28*n - 12)*x^3)*P[n - 1, x] + ((72*n + 216)*x - (44*n^3 + 216*n^2 + 340*n + 264)*x^2 + (2*n^5 + 18*n^4 + 106*n^3 + 318*n^2 + 348*n + 72)*x^3 - (2*n^5 + 18*n^4 + 62*n^3 + 102*n^2 + 80*n + 24)*x^4)*D[P[n - 1, x], x])/(36*n^2 - 36*n - (22*n^4 - 24*n^3 - 10*n^2 + 12*n)*x + (n^6 + n^5 - n^4 - n^3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((36*n^2 + 72*n - 108 - (22*n^4 + 42*n^3 - 226*n^2 - 306*n + 468)*x + (n^6 + 4*n^5 - 36*n^4 - 146*n^3 - 25*n^2 + 190*n + 12)*x^2 + (n^6 + 8*n^5 + 22*n^4 + 20*n^3 - 11*n^2 - 28*n - 12)*x^3)*v[i - 1] + ((72*n + 216)*x - (44*n^3 + 216*n^2 + 340*n + 264)*x^2 + (2*n^5 + 18*n^4 + 106*n^3 + 318*n^2 + 348*n + 72)*x^3 - (2*n^5 + 18*n^4 + 62*n^3 + 102*n^2 + 80*n + 24)*x^4)*deriv(v[i - 1], x))/(36*n^2 - 36*n - (22*n^4 - 24*n^3 - 10*n^2 + 12*n)*x + (n^6 + n^5 - n^4 - n^3)*x^2)); 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-a820c1859f68e71c","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-9985dc3bbd6a1733","e-4168e5c5dcbea788"],"text":"A178820 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A178820 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}