{"id":"n-630a3984525684cf","revision":0,"first_index":0,"fingerprint":"f6-1f45c55ca90e4ac8","name":"one_plus_derivative of basis_map_A013611 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":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["16","12"],["112","384","144"],["640","4032","6912","1728"],["3328","30720","96768","110592","20736"],["16384","199680","921600","1935360","1658880","248832"],["77824","1179648","7188480","22118400","34836480","23887872","2985984"],["360448","6537216","49545216","201277440","464486400","585252864","334430208","35831808"],["1638400","34603008","313786368","1585446912","4830658560","8918138880","9364045824","4586471424","429981696"],["7340032","176947200","1868562432","11296309248","42807066624","104342224896","160526499840","144473849856","61917364224","5159780352"],["32505856","880803840","10616832000","74742497280","338889277440","1027369598976","2086844497920","2751882854400","2167107747840","825564856320","61917364224"],["142606336","4290772992","58133053440","467140608000","2466502410240","8946676924416","22602131177472","39351924817920","45406067097600","31784246968320","10897456103424","743008370688"],["620756992","20535312384","308935655424","2790386565120","16817061888000","71035269414912","214720246185984","464958127079424","708334646722560","726497073561600","457693156343808","142657607172096","8916100448256"],["2684354560","96838090752","1601754365952","16064654082048","108825076039680","524692330905600","1846917004787712","4785194057859072","9066683478048768","12277800543191040","11333354347560960","6490921126330368","1854548893237248","106993205379072"],["11542724608","450971566080","8134399623168","89698244493312","674715471446016","3656522554933248","14691385265356800","44326008114905088","100489075215040512","169244758256910336","206267049125609472","173091230035476480","90872895768625152","23966478004912128","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","28","640","13312","262144","4980736","92274688","1677721600","30064771072","532575944704","9345848836096","162727720910848","2814749767106560","48413695994232832","828662331436171264"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2","3"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["4","24","36"],["12","36","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189099,"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-88065fba6493ff11"]},{"id":189100,"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-88065fba6493ff11"]},{"id":189101,"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-88065fba6493ff11"]},{"id":189102,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-88065fba6493ff11"]}],"region_exact":{},"incoming_edges":[{"id":"e-88065fba6493ff11","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-630a3984525684cf","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":"9f39ef11dffdc3bf0cb86e703c8b9c9642c02087576cc25b56ad030e9adb2881","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-63e0d27d213f2270","role":"sequence","name":"basis_map_A013611 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-06c7a9cc9728ef71","kind":"fingerprint_related","node_a_id":"n-0cf31549ccf2a87d","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-0cf31549ccf2a87d","other":{"id":"n-0cf31549ccf2a87d","name":"one_plus_derivative of basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["7","24","9"],["10","63","108","27"],["13","120","378","432","81"]],"card_rows":[["7","24","9"],["10","63","108","27"],["13","120","378","432","81"],["16","195","900","1890","1620","243"]],"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","D+1"],"edges":["e-33def10999a9e794","e-2d5c448bc35ca18e"],"text":"basis-map coefficients of A112227 \u2192 basis A013610 \u2192 D+1"},"headline":"basis-map coefficients of A112227 \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-13cc689e2c20548e","kind":"fingerprint_related","node_a_id":"n-33b5464f4fbbfa66","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-33b5464f4fbbfa66","other":{"id":"n-33b5464f4fbbfa66","name":"one_plus_derivative 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":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","3"],["7","24","9"]],"card_rows":[["1"],["4","3"],["7","24","9"],["10","63","108","27"]],"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","D+1"],"edges":["e-41e3c7da1a12c920","e-989277e70e66dac3"],"text":"basis-map coefficients of A026729 \u2192 basis A013610 \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-36dc21bce96934c2","kind":"fingerprint_related","node_a_id":"n-85507e03b79a9eac","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-85507e03b79a9eac","other":{"id":"n-85507e03b79a9eac","name":"one_plus_derivative of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","3"],["7","24","9"]],"card_rows":[["1"],["4","3"],["7","24","9"],["10","63","108","27"]],"row_sums_preview":["1","7","40","208","1024","4864","22528","102400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["D+1"],"edges":["e-27d4809775fa7a40"],"text":"A013610 \u2192 D+1"},"headline":"A013610 \u2192 D+1","chain":null}},{"id":"l-4f4dfd190851c4bd","kind":"fingerprint_related","node_a_id":"n-1c2a0f0db3be1b21","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-1c2a0f0db3be1b21","other":{"id":"n-1c2a0f0db3be1b21","name":"one_plus_derivative 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":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-3"],["7","24","9"]],"card_rows":[["1"],["-4","-3"],["7","24","9"],["-10","-63","-108","-27"]],"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","D+1"],"edges":["e-fe4cc1f9dd1d3096","e-77e07101b31049d4"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013610 \u2192 D+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-6730b011e5c0e8e1","kind":"fingerprint_related","node_a_id":"n-c4eea4ee44016286","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-c4eea4ee44016286","other":{"id":"n-c4eea4ee44016286","name":"one_plus_derivative of basis_map_A013610 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","3"],["14","48","18"]],"card_rows":[["1"],["4","3"],["14","48","18"],["60","378","648","162"]],"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","D+1"],"edges":["e-77bf2cb365051314","e-4644ada2b42c86dd"],"text":"A136572 \u2192 basis A013610 \u2192 D+1"},"headline":"A136572 \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-a238511a4f451e5d","kind":"fingerprint_related","node_a_id":"n-a698946f19b5cd71","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-a698946f19b5cd71","other":{"id":"n-a698946f19b5cd71","name":"one_plus_derivative of derivative of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["24","18"]],"card_rows":[["3"],["24","18"],["63","216","81"],["120","756","1296","324"]],"row_sums_preview":["0","3","42","360","2496","15360","87552","473088"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["D","D+1"],"edges":["e-5fa4095a1ef98485","e-72e08aab9a106dbe"],"text":"A013610 \u2192 D \u2192 D+1"},"headline":"A013610 \u2192 D \u2192 D+1","chain":null}},{"id":"l-b23871f6c03c199f","kind":"fingerprint_related","node_a_id":"n-1c98e5590738cedf","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-1c98e5590738cedf","other":{"id":"n-1c98e5590738cedf","name":"one_plus_derivative 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":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-3"],["7","24","9"]],"card_rows":[["1"],["-4","-3"],["7","24","9"],["-10","-63","-108","-27"]],"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","D+1"],"edges":["e-226103a223be7e17","e-f527a9e5c2f0d73d"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013610 \u2192 D+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-c70724bd1361fd7b","kind":"fingerprint_related","node_a_id":"n-2b577edb24bfe619","node_b_id":"n-630a3984525684cf","fingerprint":"f6-1f45c55ca90e4ac8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f45c55ca90e4ac8"},"other_id":"n-2b577edb24bfe619","other":{"id":"n-2b577edb24bfe619","name":"one_plus_derivative 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":"f6-1f45c55ca90e4ac8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","6"],["28","96","36"]],"card_rows":[["1"],["8","6"],["28","96","36"],["80","504","864","216"]],"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","D+1"],"edges":["e-db1885975295b772","e-a74154ba311953de"],"text":"A130123 \u2192 basis A013610 \u2192 D+1"},"headline":"A130123 \u2192 basis A013610 \u2192 D+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:      16,        12;\n  2:     112,       384,        144;\n  3:     640,      4032,       6912,        1728;\n  4:    3328,     30720,      96768,      110592,       20736;\n  5:   16384,    199680,     921600,     1935360,     1658880,       248832;\n  6:   77824,   1179648,    7188480,    22118400,    34836480,     23887872,      2985984;\n  7:  360448,   6537216,   49545216,   201277440,   464486400,    585252864,    334430208,     35831808;\n  8: 1638400,  34603008,  313786368,  1585446912,  4830658560,   8918138880,   9364045824,   4586471424,   429981696;\n  9: 7340032, 176947200, 1868562432, 11296309248, 42807066624, 104342224896, 160526499840, 144473849856, 61917364224, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy (3*n-2+3*x)*P(n,x) = (12*n+4+(36*n+24)*x+36*x^2)*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[((12*n + 4 + (36*n + 24)*x + 36*x^2)*P[n - 1, x])/(3*n - 2 + 3*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] = ((12*n + 4 + (36*n + 24)*x + 36*x^2)*v[i - 1])/(3*n - 2 + 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":["basis A013611","D+1"],"edges":["e-82b80af703a32f5e","e-88065fba6493ff11"],"text":"A013610 \u2192 basis A013611 \u2192 D+1"},"headline":"A013610 \u2192 basis A013611 \u2192 D+1","chain":null}
