{"id":"n-0f474f59a27a747e","revision":0,"first_index":0,"fingerprint":"f6-c94e3f620e523ccf","name":"theta_plus_one of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["2","8","6"],["4","32","72","64","20"],["8","96","360","640","600","288","56"],["16","256","1344","3584","5600","5376","3136","1024","144"],["32","640","4320","15360","33600","48384","47040","30720","12960","3200","352"],["64","1536","12672","56320","158400","304128","413952","405504","285120","140800","46464","9216","832"],["128","3584","34944","186368","640640","1537536","2690688","3514368","3459456","2562560","1409408","559104","151424","25088","1920"],["256","8192","92160","573440","2329600","6709248","14350336","23429120","29652480","29286400","22550528","13418496","6056960","2007040","460800","65536","4352"],["512","18432","235008","1671168","7833600","26320896","66533376","130351104","201636864","248934400","246445056","195526656","123561984","61415424","23500800","6684672","1331712","165888","9728"],["1024","40960","583680","4669440","24806400","95256576","277831680","635043840","1160939520","1719910400","2081091584","2063892480","1676912640","1111326720","595353600","254017536","84341760","21012480","3696640","409600","21504"],["2048","90112","1419264","12615680","74905600","323592192","1069651968","2794192896","5894000640","10187161600","14567641088","17336696832","17216303104","14262026240","9823334400","5588385792","2597726208","970776576","284641280","63078400","9934848","991232","47104"],["4096","196608","3391488","33161216","217620480","1044578304","3859136512","11341135872","27112402944","53555363840","88366350336","122690469888","143990898688","143138881536","120499568640","85688582144","51212316672","25517555712","10474799104","3481927680","914006016","182386688","26001408","2359296","102400"],["8192","425984","7987200","85196800","612352000","3233218560","13202309120","43109580800","115183411200","255963136000","478651064320","759512432640","1028506419200","1192824012800","1186738176000","1012683243520","739733463040","460733644800","243164979200","107773952000","39606927360","11855134720","2816819200","511180800","66560000","5537792","221184"],["16384","917504","18579456","214695936","1677312000","9661317120","43207557120","155194490880","458308730880","1131626496000","2365099376640","4221995581440","6479590440960","8588332892160","9859055616000","9815237591040","8473310576640","6332993372160","4085171650560","2263252992000","1069387038720","426784849920","141967687680","38645268480","8386560000","1395523584","167215104","12845056","475136"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","16","192","2048","20480","196608","1835008","16777216","150994944","1342177280","11811160064","103079215104","893353197568","7696581394432","65970697666560"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","-1"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["2","6","6","2"],["0","4","8","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182632,"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-9db0d93fff704495"]},{"id":182633,"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-9db0d93fff704495"]},{"id":182634,"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-9db0d93fff704495"]}],"region_exact":{},"incoming_edges":[{"id":"e-9db0d93fff704495","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-0f474f59a27a747e","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":"08583e1d98ccc59568f185ee02fe0d285a0c595d3db378b2f3536a99e665a724","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-336459bdce627a87","role":"sequence","name":"Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","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-01320f8e4e132873","kind":"fingerprint_related","node_a_id":"n-3c1634d384f562fa","node_b_id":"n-0f474f59a27a747e","fingerprint":"f6-c94e3f620e523ccf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c94e3f620e523ccf"},"other_id":"n-3c1634d384f562fa","other":{"id":"n-3c1634d384f562fa","name":"theta_plus_one of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-c94e3f620e523ccf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4","3"],["1","8","18","16","5"]],"card_rows":[["1"],["1","4","3"],["1","8","18","16","5"],["1","12","45","80","75","36","7"]],"row_sums_preview":["1","8","48","256","1280","6144","28672","131072"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["\u03b8+1"],"edges":["e-893f1bb3064921cb"],"text":"A034870 \u2192 \u03b8+1"},"headline":"A034870 \u2192 \u03b8+1","chain":null}},{"id":"l-029dd5e16b8fa675","kind":"fingerprint_related","node_a_id":"n-219cad7cbf04e6ec","node_b_id":"n-0f474f59a27a747e","fingerprint":"f6-c94e3f620e523ccf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c94e3f620e523ccf"},"other_id":"n-219cad7cbf04e6ec","other":{"id":"n-219cad7cbf04e6ec","name":"one_plus_derivative of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-c94e3f620e523ccf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","4","1"],["5","16","18","8","1"]],"card_rows":[["1"],["3","4","1"],["5","16","18","8","1"],["7","36","75","80","45","12","1"]],"row_sums_preview":["1","8","48","256","1280","6144","28672","131072"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["D+1"],"edges":["e-ef6e801bbc7a43dc"],"text":"A034870 \u2192 D+1"},"headline":"A034870 \u2192 D+1","chain":null}},{"id":"l-02b7edf3fe2716fb","kind":"fingerprint_related","node_a_id":"n-0f474f59a27a747e","node_b_id":"n-ba46e27673af7615","fingerprint":"f6-c94e3f620e523ccf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c94e3f620e523ccf"},"other_id":"n-ba46e27673af7615","other":{"id":"n-ba46e27673af7615","name":"one_plus_derivative of basis_map_A013609 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-c94e3f620e523ccf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","16","4"],["80","256","288","128","16"]],"card_rows":[["1"],["12","16","4"],["80","256","288","128","16"],["448","2304","4800","5120","2880","768","64"]],"row_sums_preview":["1","32","768","16384","327680","6291456","117440512","2147483648"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013609","D+1"],"edges":["e-9638204c7017296b","e-bd54797fd4601349"],"text":"A034870 \u2192 basis A013609 \u2192 D+1"},"headline":"A034870 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-1f455a2c4653e3af","kind":"fingerprint_related","node_a_id":"n-0f474f59a27a747e","node_b_id":"n-858b0d55747d5753","fingerprint":"f6-c94e3f620e523ccf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c94e3f620e523ccf"},"other_id":"n-858b0d55747d5753","other":{"id":"n-858b0d55747d5753","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-c94e3f620e523ccf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","32","8"],["320","1024","1152","512","64"]],"card_rows":[["1"],["24","32","8"],["320","1024","1152","512","64"],["3584","18432","38400","40960","23040","6144","512"]],"row_sums_preview":["1","64","3072","131072","5242880","201326592","7516192768","274877906944"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013609","D+1"],"edges":["e-65b163cdbb61d2ab","e-d857b9ad6c2e00bb"],"text":"A139548 \u2192 basis A013609 \u2192 D+1"},"headline":"A139548 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-952c1c43546f4849","kind":"fingerprint_related","node_a_id":"n-7bd744c0dbd95f40","node_b_id":"n-0f474f59a27a747e","fingerprint":"f6-c94e3f620e523ccf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c94e3f620e523ccf"},"other_id":"n-7bd744c0dbd95f40","other":{"id":"n-7bd744c0dbd95f40","name":"one_plus_derivative of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-c94e3f620e523ccf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","8","2"],["20","64","72","32","4"]],"card_rows":[["1"],["6","8","2"],["20","64","72","32","4"],["56","288","600","640","360","96","8"]],"row_sums_preview":["1","16","192","2048","20480","196608","1835008","16777216"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["D+1"],"edges":["e-9c1db95755a63baa"],"text":"A139548 \u2192 D+1"},"headline":"A139548 \u2192 D+1","chain":null}},{"id":"l-c1ab13761c82d987","kind":"fingerprint_related","node_a_id":"n-0f474f59a27a747e","node_b_id":"n-062516b812f0ea15","fingerprint":"f6-c94e3f620e523ccf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c94e3f620e523ccf"},"other_id":"n-062516b812f0ea15","other":{"id":"n-062516b812f0ea15","name":"theta_plus_one of basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-c94e3f620e523ccf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","32","24"],["64","512","1152","1024","320"]],"card_rows":[["1"],["8","32","24"],["64","512","1152","1024","320"],["512","6144","23040","40960","38400","18432","3584"]],"row_sums_preview":["1","64","3072","131072","5242880","201326592","7516192768","274877906944"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013609","\u03b8+1"],"edges":["e-65b163cdbb61d2ab","e-080a31bf6a6c0614"],"text":"A139548 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A139548 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-c9b590305d5bb7a8","kind":"fingerprint_related","node_a_id":"n-0f474f59a27a747e","node_b_id":"n-af8dc2447e44dda7","fingerprint":"f6-c94e3f620e523ccf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c94e3f620e523ccf"},"other_id":"n-af8dc2447e44dda7","other":{"id":"n-af8dc2447e44dda7","name":"theta_plus_one of basis_map_A013609 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-c94e3f620e523ccf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16","12"],["16","128","288","256","80"]],"card_rows":[["1"],["4","16","12"],["16","128","288","256","80"],["64","768","2880","5120","4800","2304","448"]],"row_sums_preview":["1","32","768","16384","327680","6291456","117440512","2147483648"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013609","\u03b8+1"],"edges":["e-9638204c7017296b","e-8e6d4528786be9f7"],"text":"A034870 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A034870 \u2192 basis A013609 \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:   1;\n  1:   2,     8,      6;\n  2:   4,    32,     72,      64,      20;\n  3:   8,    96,    360,     640,     600,      288,       56;\n  4:  16,   256,   1344,    3584,    5600,     5376,     3136,      1024,       144;\n  5:  32,   640,   4320,   15360,   33600,    48384,    47040,     30720,     12960,      3200,       352;\n  6:  64,  1536,  12672,   56320,  158400,   304128,   413952,    405504,    285120,    140800,     46464,      9216,       832;\n  7: 128,  3584,  34944,  186368,  640640,  1537536,  2690688,   3514368,   3459456,   2562560,   1409408,    559104,    151424,    25088,     1920;\n  8: 256,  8192,  92160,  573440, 2329600,  6709248, 14350336,  23429120,  29652480,  29286400,  22550528,  13418496,   6056960,  2007040,   460800,   65536,    4352;\n  9: 512, 18432, 235008, 1671168, 7833600, 26320896, 66533376, 130351104, 201636864, 248934400, 246445056, 195526656, 123561984, 61415424, 23500800, 6684672, 1331712, 165888, 9728;\n","recurrence_sentence":"The row polynomials satisfy (1+(2*n-1)*x)*P(n,x) = (2+(4*n+6)*x+(8*n+6)*x^2+(4*n+2)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + (4*n + 6)*x + (8*n + 6)*x^2 + (4*n + 2)*x^3)*P[n - 1, x])/(1 + (2*n - 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (4*n + 6)*x + (8*n + 6)*x^2 + (4*n + 2)*x^3)*v[i - 1])/(1 + (2*n - 1)*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-336459bdce627a87","steps":["\u03b8+1"],"edges":["e-9db0d93fff704495"],"text":"A139548 \u2192 \u03b8+1"},"headline":"A139548 \u2192 \u03b8+1","chain":null}