{"id":"n-8c5ad656fba85f85","revision":0,"first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","name":"one_plus_derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["640","672","144"],["13312","23040","12096","1728"],["262144","638976","552960","193536","20736"],["4980736","15728640","19169280","11059200","2903040","248832"],["92274688","358612992","566231040","460062720","199065600","41803776","2985984"],["1677721600","7751073792","15061745664","15854469120","9661317120","3344302080","585252864","35831808"],["30064771072","161061273600","372051542016","481975861248","380507258880","185497288704","53508833280","8026324992","429981696"],["532575944704","3246995275776","8697308774400","13393855512576","13013348253696","8218956791808","3338951196672","825564856320","108355387392","5159780352"],["9345848836096","63909113364480","194819716546560","347892350976000","401815665377280","312320358088704","164379135836160","57239163371520","12383472844800","1444738498560","61917364224"],["162727720910848","1233652046364672","4218001482055680","8572067528048640","11480447582208000","10607933565960192","6871047877951488","3099720847196160","944446195630080","181624268390400","19070548180992","743008370688"],["2814749767106560","23432791811162112","88822947338256384","202464071138672640","308594431009751040","330636890367590400","254590405583044608","141347270632144896","55794975249530880","15111139130081280","2615389464821760","249650812551168","8916100448256"],["48413695994232832","439100963668623360","1827757761270644736","4618793261589331968","7896098774408232960","9628146247504232448","8596559149557350400","5673729038707851264","2756271777326825472","967112904325201920","235733770429267968","37090977864744960","3245460563165184","106993205379072"],["828662331436171264","8133500927031115776","36884480948164362240","102354434631156105216","193989316986751942656","265308918820116627456","269588094930118508544","206317419589376409600","119148309812864876544","51450406510100742144","16247496792663392256","3600297584737910784","519273690106429440","41941336508596224","1283918464548864"],["14123288431433875456","149159219658510827520","732015083432800419840","2213068856889861734400","4605949558402024734720","6983615411523069935616","7959267564603498823680","6932265298203047362560","4642141940760969216000","2382966196257297530880","926107317181813358592","265868129334491873280","54004463771068661760","7189943401473638400","539245755110522880","15407021574586368"],["239807672958224171008","2711671378835304087552","14319285087217039441920","46848965339699226869760","106227305130713363251200","176868463042637749813248","223475693168738237939712","218311338914838824878080","166374367156873136701440","99032361402900676608000","45752950968140112592896","16164782263537105895424","4253890069351869972480","797604388003475619840","98604938077352755200","6902345665414692864","184884258895036416"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1456","50176","1668352","54089728","1721036800","53971714048","1673123135488","51381071773696","1565611481104384","47391482671267840","1426483628405161984","42728160776415084544","1274413838809597739008","37868296924628047101952","1121484178152446010327040"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["640","672","144"]],"recurrence":{"denominator":{"coeffs":[["7","3"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["160","168","36"],["48","36","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194320,"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-225d37c726225e58"]},{"id":194321,"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-225d37c726225e58"]},{"id":194322,"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-225d37c726225e58"]},{"id":194323,"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-225d37c726225e58"]}],"region_exact":{},"incoming_edges":[{"id":"e-225d37c726225e58","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-8c5ad656fba85f85","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":"f755a85474a5a7dde878cdd468d29b1813b32f7eb5b0a373008bd5d05cc880aa","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-fcf922cdd7b5dfe1","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","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-09b74977a88ab770","kind":"fingerprint_related","node_a_id":"n-4bf16590d3e16b9c","node_b_id":"n-8c5ad656fba85f85","fingerprint":"f6-87f28ed98a2aba0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-87f28ed98a2aba0a"},"other_id":"n-4bf16590d3e16b9c","other":{"id":"n-4bf16590d3e16b9c","name":"one_plus_derivative of basis_map_A007318 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","3"],["40","42","9"]],"card_rows":[["1"],["7","3"],["40","42","9"],["208","360","189","27"]],"row_sums_preview":["1","10","91","784","6517","52822","420175","3294172"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A007318","D+1"],"edges":["e-5bade392cbb9f615","e-8a776f8940c815b2"],"text":"A013610 \u2192 basis A007318 \u2192 D+1"},"headline":"A013610 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-3b04a1fdb650dd63","kind":"fingerprint_related","node_a_id":"n-e63fbd290efa0220","node_b_id":"n-8c5ad656fba85f85","fingerprint":"f6-87f28ed98a2aba0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-87f28ed98a2aba0a"},"other_id":"n-e63fbd290efa0220","other":{"id":"n-e63fbd290efa0220","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","9"],["360","378","81"]],"card_rows":[["1"],["21","9"],["360","378","81"],["5616","9720","5103","729"]],"row_sums_preview":["1","30","819","21168","527877","12835746","306307575","7204354164"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["D+1"],"edges":["e-8f0ce1d5978cf162"],"text":"A038335 \u2192 D+1"},"headline":"A038335 \u2192 D+1","chain":null}},{"id":"l-4670b257576823bb","kind":"fingerprint_related","node_a_id":"n-7dc80baa1678cc50","node_b_id":"n-8c5ad656fba85f85","fingerprint":"f6-87f28ed98a2aba0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-87f28ed98a2aba0a"},"other_id":"n-7dc80baa1678cc50","other":{"id":"n-7dc80baa1678cc50","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["160","168","36"],["1664","2880","1512","216"],["16384","39936","34560","12096","1296"]],"card_rows":[["160","168","36"],["1664","2880","1512","216"],["16384","39936","34560","12096","1296"],["155648","491520","599040","345600","90720","7776"]],"row_sums_preview":["364","6272","104272","1690304","26891200","421654016","6535637248","100353655808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["D+1"],"edges":["e-36faf349c7bd0aed"],"text":"A038284 \u2192 D+1"},"headline":"A038284 \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:             640,              672,              144;\n  1:           13312,            23040,            12096,             1728;\n  2:          262144,           638976,           552960,           193536,             20736;\n  3:         4980736,         15728640,         19169280,         11059200,           2903040,            248832;\n  4:        92274688,        358612992,        566231040,        460062720,         199065600,          41803776,          2985984;\n  5:      1677721600,       7751073792,      15061745664,      15854469120,        9661317120,        3344302080,        585252864,         35831808;\n  6:     30064771072,     161061273600,     372051542016,     481975861248,      380507258880,      185497288704,      53508833280,       8026324992,       429981696;\n  7:    532575944704,    3246995275776,    8697308774400,   13393855512576,    13013348253696,     8218956791808,    3338951196672,     825564856320,    108355387392,      5159780352;\n  8:   9345848836096,   63909113364480,  194819716546560,  347892350976000,   401815665377280,   312320358088704,  164379135836160,   57239163371520,  12383472844800,   1444738498560,    61917364224;\n  9: 162727720910848, 1233652046364672, 4218001482055680, 8572067528048640, 11480447582208000, 10607933565960192, 6871047877951488, 3099720847196160, 944446195630080, 181624268390400, 19070548180992, 743008370688;\n","recurrence_sentence":"The row polynomials satisfy (3*n+7+3*x)*P(n,x) = (48*n+160+(36*n+168)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 640 + 672*x + 144*x^2.","mathematica":"P[0, x_] := 640 + 672*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((48*n + 160 + (36*n + 168)*x + 36*x^2)*P[n - 1, x])/(3*n + 7 + 3*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 640 + 672*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48*n + 160 + (36*n + 168)*x + 36*x^2)*v[i - 1])/(3*n + 7 + 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-a93042cd70c4761e","steps":["basis A013610","D+1"],"edges":["e-77aaa200c8a95097","e-225d37c726225e58"],"text":"A038330 \u2192 basis A013610 \u2192 D+1"},"headline":"A038330 \u2192 basis A013610 \u2192 D+1","chain":null}